mysql> create table test2 (group_id int unsigned not null auto_increment,object_id int unsigned not null, primary key (group_id,object_id));
Query OK, 0 rows affected (0.04 sec)

mysql> insert into test2 values (null,1),(null,2); 
Query OK, 2 rows affected (0.00 sec)
Records: 2  Duplicates: 0  Warnings: 0

mysql> select * from test2;
+----------+-----------+
| group_id | object_id |
+----------+-----------+
|        1 |         1 | 
|        2 |         2 | 
+----------+-----------+
2 rows in set (0.00 sec)

mysql> insert into test2 values (1,2),(null,1);
Query OK, 2 rows affected (0.00 sec)
Records: 2  Duplicates: 0  Warnings: 0

mysql> select * from test2;
+----------+-----------+
| group_id | object_id |
+----------+-----------+
|        1 |         1 | 
|        1 |         2 | 
|        2 |         2 | 
|        3 |         1 | 
+----------+-----------+
4 rows in set (0.00 sec)