Symmetric groups and conjugacy classes

Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $α,β\in S_n$, we prove that the product $α^{S_n}β^{S_n}$ of the conjugacy classes $α^{S_n}$ and $β^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $α^{S_n}β^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $α,β\in S_n$ in the case when $α^{S_n}β^{S_n}$ is the union of exactly two distinct conjugacy classes.