One of the largest active facebook communities promoting a social good is the "Join to support Breast Cancer Awareness Month". Started over one year ago, this group has over 798,000 members, and continues to grow. Since its beginning the group's discussion board has hosted over 500 discussion topics. People continue to actively participate in the discussion board every few hours.
The group's purpose is stated under the group's description, "The purpose of this group is to raise awareness surrounding this issue and to think of new ways for us to contribute to the cause. Please use the discussion board and wall to discuss new ideas, issues and important campaigns related to the search for the cure. Join us to promote new ways of fighting breast cancer!" The group also has hosted Donation Drives. The main tools used by this group are person-to-person invitations, and making the group open for anyone to join or invite friends to join.
This is not everything the group could necessarily be doing, but it is both free and easy to simply let members invite their friends. This method has obviously been effective, since the group is nearly at 800,000, with the number of members growing literally every few minutes. Some of thos that join invite more friends of theirs to join as well, helping the group grow rapidly.