Let $AGL(2d,2)$ be the affine general linear group acting natrually on a $2d$-dimensional vector space over $GF(2)$. Is there a regular subgroup of $AGL(2d,2)$ isomorphic to $Z_{2^d}:Z_{2^d}$ for $d=3$ and $4$ respectively?
I tried to compute all regular subgroups of a Sylow $2$-subgroup of $AGL(2d,2)$ using the magma command "RegularSubgroups" but failed since it's too memory-consuming.