The octahedron inside a cube.

 

The cartesian directions are the C4 axes. They are also C2(=C42). The body diagonals are the C3 axes (four altogether). The line joining a pair of opposite edges of the cube is a C2 axis. There are six such axes. These pure rotational symmetry elements are illustrated in the top part of the figure, below:

The octahedron also has an inversion centre. Combining the pure rotations with i generates several other improper rotations. There are S6 axes coincident with the C3 axes and S4 axes coincident with the C4 axes. There are horizontal mirror planes normal to each C4 axis. There are diagonal mirror planes midway between pairs of C4 axis. Two of these improper rotations(horizontal and diagonal mirror planes) are illustrated in the lower half of the diagram.

Try to work out the transformation of the coorrdinate axes and the six vertices upon each one of the symmetry operations.

For example, for C2(2), we have x--> y, y-->x, z-->-z, 1-->2, 2-->1, 3-->6, 6-->3, 4-->5, 5-->4.

 

The full list of symmetry operations is given below, arranged into classes. (Remember that, a C3, C4, S4 and S6 elements generate two independent operations, each!): E, 8C3, 6C2, 6C4, 3C2, i, 6S4, 8S6, 3sh, 6sd