Example 3 A cube has 8 vertices, each inlaid with a diamond. There are $m$ different colors of diamonds available. Find the number of different inlaying methods (assuming the 8 vertices are indistinguishable).
Solve As shown in Figure 1-11, $N_{8}=\{1,2,3,4,5,6,7,8\}, N_{8}$ has a group $G=$ $\left\{P_{1}, P_{2}, \cdots, P_{24}\right\},$ and the permutations in $G$ can be divided into 5 types (Figure 1-12).
Type I. Permutations that leave the cube unchanged, such as $P_{1}=I=(1)(2)(3)(4)(5)(6)(7)$ (8), so $C\left(P_{1}\right)=8$.

Type II. Rotations of $\pm 90^{\circ}$ about the axis through the centers of opposite faces of the cube. There are 6 such permutations: $P_{2}, P_{3}, \cdots, P_{7}$. Each can be written as the product of 2 4-cycles. For example, $P_{2}=(1$ $234)(5678)$, so $C\left(P_{i}\right)=2(i=2,3, \cdots, 7)$.

Type III. Rotations of $180^{\circ}$ about the axis through the centers of opposite faces of the cube. There are 3 such permutations: $P_{8}, P_{9}, P_{10}$. Each can be written as the product of 4 2-cycles. For example, $P_{8}=(13)(24)(5$ 7) (68), so $C\left(P_{i}\right)=4(i=8,9,3)$.
Type IV. Rotations of $\pm 120^{\circ}$ about the axis through the cube's diagonals. There are 8 such permutations: $P_{11}, P_{12}, \cdots, P_{18}$. Each can be written as the product of 2 1-cycles and 2 3-cycles. For example, $P_{11}=(1)(7)(254)(368)$, so $C\left(P_{i}\right)=4(i=11,12, \cdots, 18)$.
Type V. Rotations of $180^{\circ}$ about the axis through the midpoints of opposite edges of the cube. There are 6 such permutations: $P_{19}, P_{20}, \cdots, P_{24}$. Each is the product of 4 2-cycles. For example, $P_{19}=(15)(28)(37)(46)$, so $C\left(P_{i}\right)=4(i=$ $19,20, \cdots, 124)$.
By Theorem (1), the number of different embedding methods is
$$
\begin{aligned}
M & =\frac{1}{24} \sum_{i=1}^{24} m^{c\left(P_{i}\right)}=\frac{1}{24}\left(m^{8}+6 m^{2}+3 m^{4}+8 m^{4}+6 m^{4}\right) \\
& =\frac{1}{24} m^{2}\left(m^{6}+17 m^{2}+6\right) .
\end{aligned}
$$