Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).
Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class of order 6 elements", and "6B" would be a different conjugacy class of order 6 elements; the conjugacy class 1A is the conjugacy class of the identity. In some cases, conjugacy classes can be described in a uniform way; for example, in the symmetric group they can be described by cycle structure.
The symmetric group consisting of the 6 permutations of three elements, has three conjugacy classes:
The symmetric group consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders: