Is there any routine technique to find a set of permutations which generate a Sylow 2-subgroup of the symmetric group $S\_{2^{r−1}}$? 



the symmetric group $S_{2^{r−1}}$
The Sylow $2$-subgroup of $S\_n$, where $n$ is a power of $2$, is the automorphism group of a perfect binary tree with $n$ leaves. So, by induction, you can generate the Sylow $2$-subgroup by generators of $S\_{n/2}$, plus anything which swaps the children of the root:


$\langle (1~2), \\\ (1~3)(2~4),\\\ (1~5)(2~6)(3~7)(4~8),\\\ ..., \\\ (1~~1+n/2)(2~~2+n/2)(3~~3+n/2)...(n/2~~n) \rangle$


