  For each $d\geq 3$, $n \geq 10$, and $k_1, k_2, \ldots, k_{d-1}\geq 2$ with
$k_1+k_2+\ldots+k_{d-1}\leq n-1$, we construct a regular $d$-polytope whose
automorphism group is of order $2^n$ and whose Schl\"afli type is
$\{2^{k_1},2^{k_2}, \ldots, 2^{k_{d-1}}\}$.
