Dear All, by the paper of Carter and Fong, we know the structure of 2sylow subgroups of $GL(n,q)$. Let $W_{r1}=Z_2\wr Z_2\wr...Z_2$ (r1 times), and $W$ is a 2sylow subgroup of $GL(2,q)$, then $W_r=W \wr W_{r1}$ is a 2sylow subgroup of $GL(2^r,q)$.If $n=2^{r_1}+2^{r_2}+...2^{r_k}$, then $P=W_{r_1} \times ...\times W_{r_k}$ is 2sylow subgroup of $GL(n,q)$. I need to know the permutations which generate $W_{r1}$. so by stather's paper I could have the structure of $W_r$ as a subgroup of $GL(n,q)$ (I could have the matrices which generate $W_r$). Thank you.
Derek's comment answers this question, however maybe I can add a little detail for the one aspect that is slightly tricky. It should be pretty clear how to turn a Sylow 2subgroup of $S_{2^{r1}}$ into a set of $2^r\times 2^r$block matrices with blocks of size $2$. Which means that the only (potentially) tricky thing is to write down matrices for a Sylow $2$subgroup of $GL_2(q)$. For this, there are two easy cases and a hard case:
I hope that makes sense. If you need more details, tell me. If you need an ecopy of Carter's book I'll happily share it. 


${\rm GL}_2(q)$
acting on one of the $2 \times 2$ blocks, generate a Sylow $2$−subgroup of ${\rm GL}_{2^r}(q)$. – Derek Holt Apr 18 '13 at 8:10