Apple is apparently killing Game Center as of iOS 10. Its not available in the first iOS 10 beta and isn’t available at the Apple App Store (although, interestingly, it’s still mentioned in the Settings app of the iOS 10 beta and in developer notes).
With the service, games on iOS and OS X can (could?) take advantage of Game Center, Apple’s social gaming network. It enables game app users to track their best scores on a leaderboard, compare their achievements, invite friends to play a game, and start a multiplayer game through auto-matching.
Game Center became publicly available with the release of iOS 4.1 on Sept. 8, 2010, and in November, became available in iOS 4.2 for iPad. The service underwent a major update in October 2011, for iOS 5, including support for turn-based matches.