Notation for all permutations of a set

I think you are looking for the symmetric group for which there are several notations, e.g. $\mathfrak{G}_X$ or $\mathcal S_X$.

The group of the permutations of $X$ (even if $X$ is infinite) is denoted by : $S(X)$, $\mathrm{Aut}(X)$, or $\mathfrak{S}(X)$.

If $X$ is finite with $n$ elements, it is denoted by $S_n$ or $\mathfrak S_n$.