First of all, you need to note that a user can 'belong' to several groups, because a user object can have several locations. supposing you're working only with the main location, you can do something like
$userObject = $user->attribute( 'contentobject' );
$userMainNode = $userObject->attribute( 'main_node' );
$userGroup = $userMainNode->attribute( 'parent' );
$userGroupName = $userGroup->attribute( 'name' ); // and this will be the name of the user group
once you have $userGroup, you can easily get its dataMap and play with it.
Note also you don't need all those lines. you can do it just like
/* first fetch the user content object attribute */
$user=eZUser::currentUser();
/* dump user groups array */
var_dump( $user->attribute( 'groups' ) );