Not exactly sure why your share is not working properly, especially if other items can be shared. I have the groups module and another module that i created based on the groups module and both allow things to be shared to facebook. So, that leads me to believe that you may have something wrong with your groups module itself. If you dont have any groups other than test groups, uninstall the groups module, delete the groups folder from your site, and upload a fresh copy of the groups module and try again to share to facebook.
Or, if you have a test site, try installing a fresh copy of the groups module, add a test group, and see if that shares properly.