How to add a value to the object group in group module?
Groups are fieldable entities, so you could add a field with a fieldname of, say, creation_date.

And if you'd then use the Rules module, you could create a custom rule like so:


Rules Event: After saving a new group.
Rules Action: Set a data value, i.e. to set the value of your creation_date field.


Note: If the field (like such creation_date field) does not show up within the Rules Action to be added, you may have to first add a Rules Condition like "Entity has field".