can a facebook group have a profile picture?

[Anniemack Wood]

Yes, if you go to the groups page, click on the picture next to the title of the group. It will bring up a box with the option to upload a profile picture for it.
If you go to the upper right hand area, you can also change the settings. Other cool things you can do with a facebook group are:
you can create a chat with all the members who are online at the time
you can create an event for the group
you can create a document
you can create photos and albums specifically for the group

You can also write on the groups wall like a regular wall.
I hope this helps.