TITLE: Number of colorings of cube's faces
QUESTION [7 upvotes]: How many ways are there to color faces of a cube with N colors if 
two colorings are the same if it's possible to rotate the cube such that
one coloring goes to another?

REPLY [7 votes]: The number of different colorings is equal to 
\begin{equation*}
\frac{n^6 + 3n^4 + 12n^3 + 8n^2}{24}.
\end{equation*}
You can get this number using Burnside lemma.
The wikipedia article contains solution of your problem as well.