The simple way:
g = CayleyGraph[
PermutationGroup[{Cycles[{{1, 2, 3}}], Cycles[{{1, 5}, {2, 4}}]}]]

Graph3D[g, VertexSize -> 0.4, VertexStyle -> Gray]

We can get something that looks more similar to your example by:
Flattening the colours and disabling 3D shading: use Glow
with your preferred colours and set the normal colour to Black
.
Thicken the edges. We need a new EdgeShapeFunction
for this because edges are rendered as Tube
by default, which does not respond to thickness directives.
Add "fog" with Raster3D to fade the background.
edges = EdgeList[g] //
Map[Style[#, Black,
Glow@First@PropertyValue[{g, #}, EdgeStyle]] &] //
ReplaceAll[
e : DirectedEdge[args___] /; EdgeQ[g, Reverse[e]] :>
Sort@UndirectedEdge[args]
] //
DeleteDuplicatesBy[First];
vertices = VertexList[g];
g3d = Graph3D[vertices, edges,
VertexStyle -> Directive[Black, Glow@GrayLevel[2/3]],
VertexSize -> 1/2,
EdgeShapeFunction -> (Tube[Line[#1], 0.05] &)
]
Show[g3d,
Graphics3D[
Raster3D[
ConstantArray[.05, {10, 10, 10}], {Scaled[{0, 0, 0}],
Scaled[{1, 1, 1}]}, ColorFunction -> Opacity]
]
]

Here's another one with higher resolution shading:
Show[g3d,
Graphics3D[
Raster3D[
ConstantArray[.02, 50 {1, 1, 1}], {Scaled[{0, 0, 0}],
Scaled[{1, 1, 1}]}, ColorFunction -> Opacity]
]
]

Warning: For some reason, increasing the Raster3D size from 50 to 51 disables the fog shading on my machine (M11.3, macOS 10.14.3).
At first I tried to create the corresponding polyhedron and set it to transparent white to fade the rear objects equally. The problem is that parts of them are outside the polyhedron.
It could be done like this with IGraph/M:
Needs["IGraphM`"]
Show[
g3d,
Graphics3D[
GraphicsComplex[
GraphEmbedding[g3d], {Opacity[0.7], Black, Glow[White],
Polygon[IGFaces@UndirectedGraph[g3d]]}]
]
]

Graph3D
to generate 3D-graphs. You can obtain a fog effect by overlaying the resulting graph with a semi-transparentImage3D
. See also here. $\endgroup$ – Henrik Schumacher Mar 4 '19 at 9:17Graph3D@CayleyGraph[ ..., VertexSize -> .5]
$\endgroup$ – Kuba♦ Mar 4 '19 at 10:06