RESOLVED: Found out what was going on. I have BP Gallery 1.0.9 installed on this site, and it has not been updated for quite a while (due to the requisite subscription that my client would rather not pay). After updating several plugins, I was getting an error appearing on a few of the site’s pages, so I suppressed the errors in wp-includes/class-wp-error.php. This, of course, removed the php Warning message, but it also prevented me from seeing the Fatal Error that should’ve been appearing on the individual group pages.
So, summation: 1) BP Gallery 1.0.9 may not be fully compatible with BuddyPress 1.7.2, and 2) a 404 is not always a 404. If you have suppressed your errors and get the “This page can’t be found” business, allow errors once again and re-check.