Five bodies --- non-planar CCs
There are four classes of non-planar central configurations: CC6 -- CC9. We present their approximate coordinates and pictures below. 

To check the length of segments between points (bodies) qi and qj in CC number noK, someone can write dCC[i, j, noK]

r[q_]:=Sqrt[q[[1]]^2 + q[[2]]^2 + q[[3]]^2];
d[q1_,q2_]:=r[q1-q2];
pCC[i_, points_] := points[[i+1]]; (* i-0,..,n-1*)
dCC[i_,j_, points_]:=d[points[[i+1]],points[[j+1]]]; (* i-0,..,n-1*)

CC6

(* an equilateral triangle q0q1q2 in the OYZ plane with symmetrically,with respect this plane, placed bodies q3 and q4 on OX-axis *)

no6 = {(* cc6 *)
{-0.0000000011,0.6320402959,0.0000000000},  (* q0 *)
{0.0000000034,-0.3160201531,0.5473629496},  (* q1 *)
{-0.0000000021,-0.3160201405,-0.5473629569},  (* q2 *){0.6453811921,0.0000000000,0.0000000000},  (* q3 *)
{-0.6453811923,-0.0000000023,0.0000000074}}; (* q4*)
T6=Polygon[{pCC[0, no6],pCC[1, no6],pCC[2, no6]}];
data = {};
data = Append[data, {Callout[no6[[1]],0]}]; 
data = Append[data, {Callout[no6[[2]],1]}]; 
data = Append[data, {Callout[no6[[3]],2]}]; 
data = Append[data, {Callout[no6[[4]],3]}]; 
data = Append[data, {Callout[no6[[5]],4]}]; 
data = Append[data, {Callout[{0,0,0},c]}]; 

Show[(*ListPlot3D[no10,AspectRatio->Automatic,PlotRange->Full, BoundaryStyle -> Thick, Mesh->All,PlotStyle->Directive[White,Opacity[.1],Specularity[White,40]]],*)ListPointPlot3D[data,AspectRatio->Automatic, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],T6}],
Graphics3D[{Thickness[0.0025],Dashed,Magenta, Line[{pCC[4, no6],pCC[3, no6]}]}], (* segments (4,c) and (c,2) are equal; c is the center of mass *)
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[4, no6],pCC[0, no6]}]}],
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[3, no6],pCC[0, no6]}]}],
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[4, no6],pCC[1, no6]}]}],
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[3, no6],pCC[2, no6]}]}],
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[4, no6],pCC[2, no6]}]}],
Graphics3D[{Thickness[0.0025],Red,Line[{pCC[3, no6],pCC[1, no6]}]}],

Graphics3D[{Thickness[0.0025],Blue,Line[{pCC[0, no6],pCC[2, no6]}]}],
Graphics3D[{Thickness[0.0025],Blue,Line[{pCC[1, no6],pCC[2, no6]}]}],
Graphics3D[{Thickness[0.0025],Blue,Line[{pCC[0, no6],pCC[1, no6]}]}]

](* Show *)

CC7

(* a pyramid with square basis,the square q0q3q2q4 with q1 in the summit on the symmetry line; all triangular faces are equilateral; c indicates the center of mass and this lie over the plane of the square; diagonals of the square are marked as dashed blue lines  *)
no7 = {(* cc7  *)
{0.0383924200,0.6391624769,0.0000000000},
{-0.1535696799,-0.1446220380,0.5906203403},
{0.0383924200,-0.5668514579,-0.2953101701},
{0.6403144929,0.0000000000,0.0000000000},
{-0.5635296530,0.0723110190,-0.2953101701}};
S7=Triangle[{pCC[0,no7], pCC[3,no7], pCC[2, no7], pCC[4, no7]}];
data = {};
data = Append[data, {Callout[no7[[1]],0]}]; 
data = Append[data, {Callout[no7[[2]],1]}]; 
data = Append[data, {Callout[no7[[3]],2]}]; 
data = Append[data, {Callout[no7[[4]],3]}]; 
data = Append[data, {Callout[no7[[5]],4]}]; 
data = Append[data, {Callout[{0,0,0},c]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],S7}],
Graphics3D[{Thickness[0.003], Red,Line[{pCC[1, no7],pCC[0, no7]}]}],
Graphics3D[{Thickness[0.003], Red,Line[{pCC[1, no7],pCC[3, no7]}]}],
Graphics3D[{Thickness[0.003], Red,Line[{pCC[0, no7],pCC[3, no7]}]}],
Graphics3D[Line[{pCC[1, no7],pCC[4, no7]}]],
Graphics3D[Line[{pCC[1, no7],pCC[2, no7]}]],Graphics3D[{Dashed,Red, Line[{{0,0,0},pCC[0, no7]}]}],Graphics3D[{Dashed, Red,Line[{{0,0,0},pCC[3, no7]}]}],Graphics3D[{Dashed,Red, Line[{{0,0,0},pCC[2, no7]}]}],Graphics3D[{Dashed, Red,Line[{{0,0,0},pCC[4, no7]}]}],
(* diagonals of the square *)
Graphics3D[{Dashed,Blue, Line[{pCC[4, no7], pCC[3, no7]}]}],Graphics3D[{Dashed,Blue, Line[{pCC[0, no7],pCC[2, no7]}]}]

](* Show *)

CC8

(* regular tetrahedron q0q1q3q4 with body at the origin q2 *)
no8 = {(* cc8 *)
{-0.2422221032,0.6851075669,0.0000000000},
{-0.2422221032,-0.3425537835,0.5933205573},
{0.0000000000,-0.0000000000,0.0000000000},
{0.7266663096,0.0000000000,0.0000000000},
{-0.2422221032,-0.3425537835,-0.5933205573}};

T81=Polygon[{pCC[0, no8],pCC[1, no8],pCC[3, no8]}];
T82=Polygon[{pCC[1, no8],pCC[4, no8],pCC[3, no8]}];
T83=Polygon[{pCC[0, no8],pCC[4, no8],pCC[3, no8]}];
data = {};
data = Append[data, {Callout[no8[[1]],0]}]; 
data = Append[data, {Callout[no8[[2]],1]}]; 
data = Append[data, {Callout[no8[[3]],2]}]; 
data = Append[data, {Callout[no8[[4]],3]}]; 
data = Append[data, {Callout[no8[[5]],4]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],T81}],
Graphics3D[{Green,Opacity[.1],T82}],
Graphics3D[{Red,Opacity[.1],T83}],

Graphics3D[Line[{pCC[0, no8],pCC[1, no8]}]],
Graphics3D[Line[{pCC[0, no8],pCC[3, no8]}]],
Graphics3D[Line[{pCC[1, no8],pCC[3, no8]}]],

Graphics3D[Line[{pCC[4, no8],pCC[1, no8]}]],
Graphics3D[Line[{pCC[4, no8],pCC[3, no8]}]],
Graphics3D[{Line[{pCC[4, no8],pCC[0, no8]}]}],
(* dashed lines *)
Graphics3D[{Dashed, Thickness[0.003],Line[{{0,0,0},pCC[0, no8]}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{pCC[4, no8],{0,0,0}}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{{0,0,0},pCC[3, no8]}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{{0,0,0},pCC[1, no8]}]}]
](* Show *)

CC9

(* triangle pyramid ("perturbed tetrahedron") with an equilateral triangle q0q1q4 as the base and q3 at the summit; q2 is inside; c is at the origin *)
no9 ={(* cc9  *)
{-0.2959953079,0.6405820512,0.0000000000},
{-0.2959953079,-0.3202910256,0.5547603295},
{0.1100591455,0.0000000000,0.0000000000},
{0.7779267783,0.0000000000,0.0000000000},
{-0.2959953079,-0.3202910256,-0.5547603295}};
 
T91=Polygon[{pCC[0, no9],pCC[1, no9],pCC[4, no9]}];
T92=Polygon[{pCC[1, no9],pCC[4, no9],pCC[3, no9]}];
T93=Polygon[{pCC[0, no9],pCC[4, no9],pCC[3, no9]}];

data = {};
data = Append[data, {Callout[no9[[1]],0]}]; 
data = Append[data, {Callout[no9[[2]],1]}]; 
data = Append[data, {Callout[no9[[3]],2]}]; 
data = Append[data, {Callout[no9[[4]],3]}]; 
data = Append[data, {Callout[no9[[5]],4]}]; 
data = Append[data, {Callout[{0,0,0},c]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],
Graphics3D[{Green,Opacity[.1],T91}],
Graphics3D[{Blue,Opacity[.1],T92}],
Graphics3D[{Red,Opacity[.1],T93}],
(* dashed lines *)
Graphics3D[{Dashed, Thickness[0.003],Line[{pCC[2, no9],pCC[0, no9]}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{pCC[2, no9], pCC[4, no8]}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{pCC[2, no9],pCC[3, no9]}]}],
Graphics3D[{Dashed, Thickness[0.003],Line[{pCC[2, no9],pCC[1, no9]}]}]
](* Show *)