Consider two colorings the same, if each color has the same collection of neighboring colors. I've seen Polya enumeration theorem on how to color a cube with up to $m$ colors, but that doesn't work here because I want to use exactly 6 colors. I like to think the answer is 15 because once I have two adjacent colors fixed, the rest of the cube colors itself.
I want to know if there is a generalized way to count the number of colorings for a graph with $n$ verticies with exactly $n$ colors. (perhaps to Polya enumeration theory) BTW the question was originally about coloring vertex of an octohedron.