Petrie polygon projection of a hypercube graph

I am trying to generate the Petrie polygon projection of the 8-cube in Mathematica, like this one:

I tried using, for example, the HypercubeGraph function:

HypercubeGraph[8]

which generates

As you can tell from the lack of symmetry (notice the shape of the innermost ring, for example), this is not the right projection. I also tried looking at some GraphPlot options like Method:

GraphPlot[HypercubeGraph[8], Method -> "SpringElectricalEmbedding"]

but to no avail. How can I generate the Petrie polygon projection of an 8-cube in Mathematica?

SetProperty[GraphData[{"Hypercube", 8}], VertexStyle -> Red]