Check out the files in mod/groups/actions directory for 1.7 and mod/groups/actions/groups/ for 1.8. Another way of finding the action name is to look at the add a group page's HTML and find the form action. You can then find the action file by using the inspect option on the developer's plugin or manually looking for elgg_register_action() like you did.
Actions are sometimes reused, so in this case the "edit" action is the one doing the saving for new groups, too.
FYI: In most of 1.8's plugins the action for both add and edit is called "save." Looks like groups didn't get this change for some reason.
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