Example 4-14 Use two colors to color the 8 vertices of a cube, how many different schemes are there?
Similar to Example 4-13, but changing the coloring of 6 faces to the coloring of 8 vertices, as shown in Figure 4-16. The group of vertex motions that make the cube coincide is:
(1) The identity element (1)(2)(3)(4)(5)(6)(7)(8), in the format $(1)^{8}$.
(2) The permutations for rotating $\pm 90^{\circ}$ around the $x x^{\prime}$ axis are (1234)(5678), (4321)(8765), in the format (4) ${ }^{2}$, with a total of 6 conjugate classes of the same format.
(3) The permutation for rotating $180^{\circ}$ around the $x x^{\prime}$ axis is (13)(24)(57)(68), in the format (2)${}^{4}$, with 3 permutations of the same type.
(4) The permutation for rotating $180^{\circ}$ around the $y y^{\prime}$ axis is (17)(26)(35)(48), in the format (2)${}^{4}$, with 6 permutations of the same type.
(5) The permutations for rotating $\pm 120^{\circ}$ around the $z z^{\prime}$ axis are (136)(475)(8)(2), (631)(574)(2)(8), in the format $(3)^{2}(1)^{2}$, with 8 permutations of the same type.
According to Pólya's theorem, the number of different schemes is
$$
m=\frac{1}{24} \times\left(2^{8}+6 \cdot 2^{2}+3 \cdot 2^{4}+6 \cdot 2^{4}+8 \cdot 2^{4}\right)=\frac{552}{24}=23
$$