Database of finite presentations of used groups

GAP has the following:

• Finitely Presented Groups
• Presentations and Tietze Transformations

An overview of all data libraries in GAP can be found here.

The following was a comment, but should be part of the answer:

• There was a lovely book that listed lots of this sort of material presentations, Group Tables by A. D. Thomas, G. V. Wood. It may not answer your question but if you can get your hands on a copy is an interesting perspective. It was published in the Shiva Mathematics series.

Related to your question, perhaps... there's the Atlas of finite group representations. Although the atlas itself might not be what you're after, the above page contains links that might be more relevant.

In my dissertation I developed an algorithm for finding bounds on $H_2$ of a finitely presented group with finite field coefficients. I was motivated by a conjecture Quillen on the (co)homology of linear groups. As such, I included an appendix with presentations of several linear groups and the homology calculations using my algorithms. I didn't include the list of presentations for publication, but if these types of groups are of interest I could get it to you.