An octahedron consists of eight triangular faces joined at six vertices. Its dual, the cube, consists of six square faces joined at eight vertices. Being dual polyhedra, an octahedron can be formed by placing a vertex at the center of each face of a cube, and similarly a cube can be formed from an octahedron.
While the symmetry group is commmonly referred to as Octahedral Symmetry, the dualism of the cube and octahedra means that any symmetry of a cube is identical to a symmetry of a regular octahedron. As such I will demonstrate that the group of rotational symmetries of the cube is isomorphic to the permutation group S(4) and it follows that regular octahedra have the same symmetry group.
The "building block" I have chosen for the cube is the set of four central diagonals. I claim that any permutation of the diagonals is equivalent to a rotation of the cube. If all possible permutations represent rotations, then the group of rotational symmetries must be isomporphic to S(4).
The drawing to the left shows a possible numbering system for the vertices of the cube. For ease of notation, I will refer to each of the diagonals by the number of the smaller vertex it connects. This does not mean that vertex 1 will be mapped to a particular spot, but that vertex 1 or 7 could be there.
Using the numbering scheme shown, vertex 1 is connected by edges to vertices 2, 5, and 4, counterclockwise. Vertex 7 is connected to 3, 8, and 6. Any rotations will preserve this, so we must show that permuting the diagonals can preserve this orientation.
Here we transpose two diagonals 1 and 2. Note that the vertices 3 and 5 have changed position, as have 4 and 6, but the diagonals connecting them are in the same spot. It was necessary to reverse the orientation of these diagonals to represent a rotation of the original cube, but the diagonal is still in the same position, so we consider it fixed by the permutation. Note that vertex 1 is connected to vertices 2, 5, and 4, and vertex 7 is connected to 3, 8 and 6. This would not have been the case if the diagonals had maintained their orientation.
Here diagonals 1 and 3 are transposed. Again, the orientation of diagonals 2 and 4 is altered so that the cube represents a rotation, but they maintain their position. Vertex 1 is still connected to vertices 2, 5, and 4, and vertex 7 is connected to 3, 8 and 6.
Having demonstrated these two examples, I hope it is clear that since any two diagonals share a common edge, any transposition of the diagonals will represent a rotation. Furthermore, any combination of rotations is also a rotation, so any permutation of the four diagonals is equivalent to a rotation of the cube. There are 24 rotational symmetries of the cube (or regular octahedron), which is equal to the number of permutations of four elements, therefore the group of rotational symmetries of the cube (or octahedron) is isomorphic to S(4).