The group is one of the three Abelian groups of order 8 (the other two groups are non-Abelian). An example is the modulo multiplication group (which is the only modulo multiplication group isomorphic to ).

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

Its multiplication table is illustrated above.

Each element is in its own conjugacy class. The subgroups are given by , , , , , , , , , , , , , , , and . Since the group is Abelian, all of these are normal subgroups.