Conjugacy class of transpositions is the unique smallest conjugacy class of involutions

Statement
Let $$n \ge 7$$. The conjugacy class of fact about::transpositions in the fact about::symmetric group on a set of size $$n$$, is the fact about::conjugacy class of smallest size among all conjugacy classes of involutions (elements of order two). Moreover, there is no other conjugacy class of the same size.

This fails for $$n = 6$$, where the conjugacy class of triple transpositions has the same size. For $$n = 3$$ it is the only conjugacy class of involutions, while for $$n = 4,5$$, the conjugacy class of double transpositions is strictly smaller in size.