All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2_G_2(q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, An and Sn have presentations with 3 generators; 7 relations and bit-length O(log n), while SL(n,q) has a presentation with 6 generators, 25 relations and bit-length O(log n + log q).
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Alexander Lubotzky, Robert M. Guralnick, William M. Kantor, Martin Kassabov, Presentations of finite simple groups: a computational approach. J. Eur. Math. Soc. 13 (2011), no. 2, pp. 391–458DOI 10.4171/JEMS/257