A complete set of mutually conjugate group elements. Each element in a group belongs to exactly
one class, and the identity element () is always in its own class. The conjugacy class orders of all classes must be
integral factors of the group
order of the group. From the last two statements, a
group of prime order has
one class for each element. More generally, in an Abelian
group, each element is in a conjugacy class by itself.

Two operations belong to the same class when one may be replaced by the other in a new coordinate system which is accessible
by a symmetry operation (Cotton 1990, p. 52). These sets correspond directly
to the sets of equivalent operations.

is always in a conjugacy class of
its own. To find another conjugacy class take some element, say , and find the results of all similarity transformations on . For example, for , the product of by can be read of as the element at the intersection of the row
containing
(the first multiplicand) with the column containing (the second multiplicand), giving . Now, we want to find where , so pre-multiply both sides by to obtain , so is the element whose column intersects
row
in 1, i.e., .
Thus, .
Similarly, ,
and continuing the process for all elements gives

(1)

(2)

(3)

(4)

(5)

The possible outcomes are , , or , so forms a conjugacy class. To find the next conjugacy
class, take one of the elements not belonging to an existing class, say . Applying a similarity transformation gives