is one of the three Abelian groups of group order 8 (the other two being non-Abelian). Examples include the modulo multiplication groups , , , and (and no others).

The elements of this group satisfy , where 1 is the identity element, and four of the elements satisfy . The cycle graph is shown above.

Its multiplication table is illustrated above.

Since the group is Abelian, each element is in its own conjugacy class.

The subgroups are , , , , , , , and , A, B, C, D, E, F, . Since the group is Abelian, all of these are normal.