Need to store group in db - what should i store to identify group?

There's no very easy way to do both.

A group exists in an environment primarily identified by its environment-specific ID (which is just an integer).  There is also a UUID that underlies the group, but this is not easy to use (iirc there's no direct way to programatically get a group Object/ID matching a UUID if you have one to start with).

For storing a group in the DB that means the only really good option is storing the Group ID.  You can deploy that group to another environment just fine but you *must* build around the fact that its Group ID will almost always be different in other environments.