Let $G$ be the rotation group of a cube $Q.$ How many ways (up to symmetry under the action of $G$) can we colour the $8$ vertices of $Q$ using $2$ colours?
I know the Burnside or Cauchy-Frobenius Lemma, which says that for a finite group $G$ that acts on a set $S, |G| |S/G| = \sum_{a\in G}|Fix(a)|,$ where $Fix(a) := \{x \in S : ax = x\}.$ I know there are $24$ elements in $G$. Let $S$ be the number of colourings without considering symmetry. Then $|S| = 2^8.$ The rotation group induces a natural action on the set $S$, but I'm not really sure how to formally describe this action. However, once I can formally describe the $24$ elements of $G$ and then determine their corresponding $Fix$ cardinalities (the corresponding $Fix$ cardinality of element $A\in G$ is $|Fix(A)|$) it should be fine. For instance, $|Fix(I)| = 2^8,$ where $I$ is the identity rotation.
Could someone help me uncover the cardinalities of $Fix(A)$ for $A\in G$?