I recently had to try to determine the SharePoint Default Members Group of a particular site. In Site Actions->Site Permissions, you can go to a group and then click Make Default Group. What's bizarre is that there is no indication which group is default, at least not in the UI. The only way I could find is in PowerShell as follows:
$web = Get-SPWeb <myurl>
or you can get the ID through the property bag at
Not sure which way is better, but either works for me so far. It would also be nice to see it in the UI.