For each , the centralizer of in is the subgroup
Define a relation on by such that Then is an equivalence relation, and so partitions into equivalence classes, called conjugacy classes. The conjugacy class containing is the set The number of elements in this conjugacy class is equal to the number of left cosets of in because the mapping is a bijection.
Note that the conjugacy class containing the identity is always If there are distinct conjugacy classes not containing the identity, let be representatives from each of them. Then
If there is no such that then the RHS would be odd since divides 64 and so must be a power of 2. It follows that there must be some such that
is not trivial