You need 2 tables:
GROUPS - GroupId, GroupName
USERS - UserId, UserName, Password, GroupId
The GroupId in USERS links user entries to GROUPS. You can (should) also impose the link using a foreign key.
If a user can belong to more than one group, you will need a many-to-many relationship. This is achieved with a 3rd table that looks like this:
GROUPS - GroupId, GroupName
USERS - UserId, UserName, Password
USERS_GROUPS - UserId, GroupId