Currently, only users can create and moderate Facebook Groups. But, it’s looking like Facebook are working on letting Pages create and manage groups.
On a Facebook page, it looks like it’s possible to add a Groups module to the favorites at the top of the page. However, nothing obvious happens after clicking groups. Often, a bug like this on Facebook points to new projects being worked on.

After trying to add the Groups module, I found an unfinished Groups page.
Since there’s no obvious way to create a new group, I headed to the Graph API Explorer to what can be done through the API.
Creating a Group
Using a page access token, you can POST to /me/groups with a name parameter set to your new group name.
This returns your new group ID. Query that, and you’ll get full details:
{
"id": "530648920292865",
"owner": {
"category": "Website",
"name": "Cut Out + Keep",
"id": "6308333870"
},
"name": "Test",
"venue": {
"street": ""
},
"privacy": "CLOSED",
"icon": "https://fbstatic-a.akamaihd.net/rsrc.php/v2/yI/r/0pjqWL1NfkE.png",
"updated_time": "2013-01-16T01:55:46+0000",
"email": "...@groups.facebook.com",
"version": 1
}
Note that the owner is your Page!
Now, you can visit the group. Some admin actions require you to switch to use Facebook as your page.