Groups of Order 2048 with Three Generators and Three Relations

TL;DR

This paper classifies all 3-generator, 2-group groups of order 2048 with trivial Schur multiplier and provides explicit presentations for many, demonstrating they have deficiency zero.

Contribution

It explicitly enumerates and presents 3-generator 2-groups of order 2048 with trivial Schur multiplier, establishing their deficiency zero.

Findings

Exactly 78 such groups exist.

Explicit 3-generator, 3-relation presentations for 48 groups.

All these groups have deficiency zero.

Abstract

It is shown that there are exactly seventy-eight 3-generator 2- groups of order 2^11 with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.