The group of all permutations of a set M, denoted $S(M)$, where the group operation is Function composition. A Permutation is just a bijection from the set to itsef.

https://en.wikipedia.org/wiki/Symmetric_group

Cycle decomposition

An important special case is a cycle. All disjoint cycles commute (disjoint refers to the fact that they cycle different elements). Theorem: All elements in the symmetric group can be written as a product of disjoint cycles. If the union of the supports of the cycles equals the $\{1...n\}$, the decomposition is unique and we call it the Cycle decomposition.

The Conjugacy class of an element in symmetric group is determined by the length of the cycles in the cycle decomposition. Because of this the conjugacy classes are in 1-to-1 correspondence to the Partitions of $\{1...n\}$