Extraspecial 2-groups and mod-2 cohomology lengths

Let G0 (respectively G1,G2) be the extraspecial 2-group of real (respectively complex, quaternion) type of order 22n+3 (respectively 22n+2,22n+1). The purpose of this paper is to investigate the cohomology lengths chl(Gi) with i=1,2 (the case i=0 was settled up by Yalçın). We prove that chl(G0) chl(G1) chl(G2)>2n, and the equalities chl(G0)=chl(G1)=chl(G2) hold for n 3. This disproves a conjecture of Yalçın telling that chl(Gi)=s(Gi), i=1,2, with s(Gi) the minimum cardinality of a representing set of Gi.