The Group module for Drupal 7 is based around the Entity API. Individual groups have a "Group Type" in the module, which is an Entity. Since it's an Entity, it has fields.

It also has specific permissions and roles, which apply to any Group of this Group type.

The basic permissions are for anonymous, outsider, member, and administrator, but you can create unlimited Group roles of your own and set the permissions for them. The permissions control content access and creation, as well as group administration.

The main entity connected to Group is the node entity and the permissions for the different nodes on the site are easily controlled through the Group Type.