Algorithm to compute centraliser

Question

Given : $ G = \langle F \rangle \le S_n$ and $h \in G$

Find : $C_G(h) = \langle C \rangle$

Brute Force Algorithm :

1. Genrate all elements of $ \sigma \in G$
2. Check $\sigma h = h \sigma$
3. if above condition is satisfied then $K = K \cup\sigma$
4. $S = \langle K \rangle$

Algorithm :

1. For all $\sigma \in F$
2. Check $\sigma h = h \sigma$
3. if above condition is satisfied then $S = S \cup\sigma$

Question 1: Is $S$ a generating set of $C_G(h)$? How to prove it?

Question 2 : Is there any better algorithm than above ones?

Answer 1

The second algorithm doesn't work in $S_3$ for $h=(123)$ and $F=\{(12), (13)\}$