G-l is for principal resident representative, family and staff (of any rank, including clerical and custodial employees, so long as they are assigned on “resident basis”).
G-1s are for members of a permanent mission of a recognized government. 9 FAM 41.24 N.1 . The foreign government has U.S. de jure recognition.
If G-1 to G-4 seek entry to go to UN or any other international organization they are only subject to certain security grounds of inadmissibility INA §212(a)(3)(A) -(C).
If they are inadmissible on these grounds they may seek an advisory opinion to permit admission. 9 FAM 41.24 N.2.3 . G-l–G-4 are in duration of status so long as DOS recognizes credentials. 8 C.F.R. §214.2(g)(1).