Besides the already accepted answer, another way of looking at this family of groups is as follows:  

Let $V_d = (\mathbb Z/2)^d$.  General theory says that there is a universal central extension
$$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$
and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper in Advances, and I had fun with these also in a 2007 Advances paper.