What is the center of $Z_2\wr Z_2\wr\cdot\cdot\cdot\wr Z_2$ which is the wreath product of $Z_2$ $r$ times, here $Z_2$ denotes a cyclic group of order 2?
When $r=2$, I know that $Z_2\wr Z_2\cong D_4$ which is a Sylow 2-subgroup of $S_4$ and $Z(Z_2\wr Z_2)=Z_2$.
I also know that in general $Z_2\wr Z_2\wr\cdot\cdot\cdot\wr Z_2$ is a Sylow 2-subgroup of $S_{2^r}$, I wonder if $Z(Z_2\wr Z_2\wr\cdot\cdot\cdot\wr Z_2)=Z_2$ for any $r\geq2$?