Let $ M $ be a finite group of order $2^{a+1} $ and let $ M $ have a normal subgroup $ R $ such that $|M: R|=2 $. Also we know that $ R $ is an elementary abelian subgroup of $ M $. 


For example $ D\_8 $ is an example for it. Is it true that for other cases $ M $ is abelian or what information is available for $ M $?


Thanks for your help



the structure of a 2-group
The elementary Abelian $2$-group $R$ is a ${\rm GF}(2)M/R$-module of dimension $a$. As such it is a direct sum of $t$ copies of the trivial module and $\frac{a-t}{2}$-copies of the free module of rank $1$. Then $Z(M)$ has order $2^{t},$ and there are examples of $M$ and $R$ for which every possibility of $t$ between $1$ and $a$ occur, as Arturo comments. 


