Centralizers in the Symmetric Group
For two different elements $\sigma, \tau\in S_n$ which are non-conjugate it can happen that their centralizers are same (or conjugate subgroups). I learned of this just recently. I worked out and found examples: For the two partitions $n = (n-2) +2$ and $n= (n-2) + 1 + 1$ taking taking elements of the respective conjugacy classes, $\sigma = (1,2,\ldots, n-2)(n-1, n)$ and $\tau =(1,2,\ldots,n-2)$, the centralizers are both same, namely the subgroup generated by $\sigma$.
Is there are any other pair with the same behaviour in $S_n$? For a general group $G$, it is easily seen that if $z\in Z(G)$, then $g$ and $zg$ have the same centralizers. What about groups with trivial centre?