This discussion: https://web.archive.org/web/19990202074648/http://www.math.niu.edu/~rusin/known-math/95/rubik seems to culminate in a presentation (due to Dan Hoey). I did not read it carefully, I must admit. The presentation is quite complicated. For the 2x2x2 group there is this:

http://cubezzz.dyndns.org/drupal/?q=node/view/177