At this time, no. More group types are planned in a later GJ release. The type changes planned are as follows.
Open = As is
Approval = As is except content is hidden until join request approved (group it self is visible)
Private = The current Invite
Invite = The same as Approval except content only visible after joining from invite (group it self is visible)
Page = All group functionality except no user management (can't join a page)
Closed = No longer allows joins or new content and is visible
Best I can suggest at this time is use the Approval type. The group owner, group admins, or group moderators would need to approve each request to join the group, but it's the closest to what you're wanting at this time.
Kyle (Krileon)
Community Builder Team Member
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