Example 7.6 Using $m$ colors to paint the 8 vertices of a cube $V$, with each vertex painted one color, find the number of distinct vertex-colored cubes $h(m)$.
Solution: Let $(D, *)$ be the rotation group of the cube $V$, and $(G, \circ)$ be the group of vertex-colored cubes derived from $(D, *)$. Then $d V_{1}=V_{1}$ if and only if the vertices of $V_{1}$ in the same cycle of $g$ have the same color. Since there are $m$ ways to color the vertices in a cycle the same, when the type of $g$ is $\left(b_{1}, b_{2}\right)$,
$$
P_{G}\left(x_{1}, x_{2}, \cdots, x_{8}\right)=\frac{1}{24}\left(x_{1}^{8}+8 x_{1}^{2} x_{3}^{2}+9 x_{2}^{4}+6 x_{4}^{2}\right) \text {, }
$$

By Burnside's Lemma, we have
$$
\begin{aligned}
h(m) & =\frac{1}{24}\left(m^{8}+8 m^{2+2}+9 m^{4}+6 m^{2}\right) \\
& =\frac{1}{24}\left(m^{8}+17 m^{4}+6 m^{2}\right)
\end{aligned}
$$