CMSModules/Groups/Controls/GroupProfile.ascx

This control has a bunch of boolean flags, private variables with public getters and setters when seem to exist to set the visibility of the tabs in this control. As far as I can tell, none of these are used to actually control whether these are shown or not.

for example, editing WebParts\Community\GroupProfile.ascx (which uses the above control) as follows:
<cms:GroupProfile ID="groupProfileElem" runat="server" ShowMediaTab="false" />

does not hide the media Tab.