June 08, 2012, 11:54 AM — Here's how you can get the iOS Game Center to work if you are behind a firewall.
1. Contact your network administrator.
2. Inform him or her that port forwarding must be enabled for the following ports:
443 (TCP), 3478-3497 (UDP), 4398 (UDP), 5223 (TCP), 16384-16387 (UDP), and 16393-16472 (UDP).
For more, see the original article at the link below.