*Almost nothing* can be said about a group just from a presentation.  (In fact, the abelianisation is just about the only thing one can reliably compute.)  Most strikingly, there is no algorithm to recognise whether a given presentation represents the trivial group.  More generally, one cannot in general solve 'the word problem' - ie, there is no algorithm to determine whether a given element is non-trivial.

On the other hand, there is a growing realisation that, surprisingly, if one is given a solution to the word problem (by an oracle, say) then one can compute quite a lot of information.  [Daniel Groves and I][1] proved that, in these circumstances, one can determine whether the group in question is free.  [Nicholas Touikan][2] generalised this to show that one cam compute the [Grushko decomposition][3].


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=groves&s5=wilton&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq
  [2]: http://arxiv.org/abs/0906.3902
  [3]: http://en.wikipedia.org/wiki/Grushko_theorem#Grushko_decomposition_theorem