See knobbygb's explaination of the difference between a school and club.
A group is generally formed around the operation of a single aircraft (but some groups do operate more than one a/c), will generally have less members than a club, and the members often have to make a significant capital investment to join the group. The aim of the group is unlikely to be primary training, so while you can learn to fly in a group aircraft (subject to various caveats), my feeling is that most people looking for training are better off going to a school (or club) geared up for it.