Why your Facebook group might not be listed
In order to connect a Facebook Group to your Buffer account, you will need to be either the owner of the Group or an Admin of the Group. Unfortunately it's not possible to connect Facebook Groups that you are only a member of.
A note on Facebook Groups owned by Pages
Facebook now support the creation of Groups from Pages (like this). If you're looking to connect a Group to Buffer, which is owned by a Page, you'll need to make sure that your personal Facebook Profile is set up as an Admin. You can do this by following the steps below.
Visit your Facebook Page and click on the Groups option on the left hand side. From the Groups page, click on the name of your Group.
On the right hand side of your Group, add your personal Facebook Profile as a member. Once you're a member of the Group, click on the Members link, just below the search box.
Locate your profile within the Members list and click on the settings icon. From the drop down menu, click Made Admin.