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A balanced presentation for a group is a presentation with an equal number of generators and relations. A perfect group is a group that is equal to its commutator group.

I am wondering what finite perfect groups are known to admit balanced presentations? I only know two example: the trivial group and the binary icosahedral group $\langle x,y \mid x^2=y^3=(xy)^5\rangle$. What are some other examples?

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    $\begingroup$ You can find some here there's a nice one for the covering group of $M_{22}$. $\endgroup$ – Derek Holt Jun 24 at 20:29
  • $\begingroup$ It seems to be a very recent theorem that every non-positive deficiency is realised as the presentation of some finite group. The reference is Giles Gardam, Finite groups of arbitrary deficiency Bull. Lond. Math. Soc. 49 (2017), 1100–1104, arXiv. (This isn't applicable to the question here as the groups are $p$-groups, and so not perfect. But I thought it was interesting enough to mention :-) ) $\endgroup$ – user1729 Jun 25 at 15:54
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For more examples see:

C. Campbell, E. Robertson, A deficiency zero presentation for $SL(2,p)$. Bull. London Math. Soc. 12 (1980), no. 1, 17–20.

and, more recently,

C. Campbell, G. Havas, C. Ramsay, E. Robertson, Nice efficient presentations for all small simple groups and their covers. LMS J. Comput. Math. 7 (2004), 266–283.

C. Campbell, G. Havas, C. Ramsay, E. Robertson, All simple groups with order from 1 million to 5 million are efficient. Int. J. Group Theory 3 (2014), no. 1, 17–30.

Here a superperfect group (a perfect group with trivial Schur multiplier) is efficient if and only if it admits a balanced presentation.

The last two papers cover all groups of order up to 5 million.

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