The iPhone 3G at the moment doesn’t (officially) run the Game Center available on iOS 4.1. This limitation is overcome by the jailbreaking community. The jailbreak people have managed to get the Game Center running on the iPhone 3G that has been jailbroken by the RedSn0w 0.9.6 jailbreak for iOS 4.1.
The video below demonstrates that the iPhone 3G is capable of running Game Center:
