Example 7.7 Using $m$ colors to paint the 6 faces of a cube $V$, with each face painted one color, find the number of distinct face-colored cubes $g(m)$.
Let $(D, *)$ be the rotation group of the cube $V$, and $\left(F, 0^{\circ}\right)$ be the face permutation group induced by $(D, *)$. Suppose $d \in D$, and the face permutation induced by $d$ is $f$. Let $V_{1}$ be the cube obtained by coloring the 6 faces of $V$. Then $d V_{1}=V_{1}$ if and only if the faces in the same cycle of $f$ have the same color. Since there are $m$ ways to color the faces in a cycle the same, when the cycle type of $f$ is $\left(b_{1}, b_{2}, \cdots, b_{6}\right)$,
$$
P_{F}\left(x_{1}, x_{2}, \cdots, x_{6}\right)=\frac{1}{24}\left(x_{1}^{6}+8 x_{3}^{2}+6 x_{2}^{3}+3 x_{1}^{2} x_{2}^{2}+6 x_{1}^{2} x_{4}\right) \text {, }
$$

By Burnside's Lemma, we have
$$
\begin{aligned}
g(m) & =\frac{1}{24}\left(m^{6}+8 m^{2}+6 m^{3}+3 m^{2+2}+6 m^{2+1}\right) \\
& =\frac{1}{24}\left(m^{6}+3 m^{4}+12 m^{3}+8 m^{2}\right)
\end{aligned}
$$