Stabilizer of $1$ on $S_6$

Question

$S_6$ acts on the set $\{1, 2, 3, 4, 5, 6\}$ by permutation. Explicitly describe $(S_6)1$ the stabilizer of $1$ in $S_6$.

I understand that I want to list all the elements that fix $1$ in the permutation, such as $(2\ 3)$ or $(2\ 3\ 4)$, but does this mean I need to list every permutation that would occur and test them? Wouldn't that amount to $720$ different permutations that would need to be tested? I feel like I'm going about this the wrong way.

Answer 1

The group of permutation that fixes one element you can think it as a group $G\subset S_6$ such that is $G\cong S_5$.

Answer 2

$S_5$ acts on $\{2,3,4,5,6\}$ by permutation, so your answer is $\{(1)a:a\in S_5\}$