If $\sigma\in S_n$ is of length $n$, then $\sigma$ generates its centralizer

Question

Let $S_n$ and $C_G(\sigma)$ denote the symmetric goup and the centralizer of $\sigma\in S_n$, respectively. I want to show: If $\sigma$ is of length $n$, then $C_G(\sigma)=\langle\sigma\rangle$, i.e. $\sigma$ generates $C_G(\sigma)$

Answer 1

We have $\langle \sigma \rangle \subset C_G(\sigma)$, so it is enough to show that they are the same size.

But how can we quickly compute the size of $C_G(\sigma)$? The index of the centralizer is just the size of the conjugacy class of $\sigma$, or the number of $n$-cycles in $S_n$.