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

Hence

for all

is not trivial