Following the release of iOS 6, Apple just pushed an update to OS X Mountain Lion as well, which reaches version 10.8.2. Available for download on the Mac App Store, 10.8.2 brings Facebook integration to OS X, working similarly to Facebook in iOS 6, and various stability improvements. Like in iOS 6, Facebook is now available as a social service to log in from the System Preferences, and there are options to update contact information and pictures through Facebook as well as system-wide sharing menus.

On the feature set side, Facebook is now also integrated in Game Center (for friend recommendations and like buttons for games), Power Nap is now supported on the late 2010 MacBook Air, and iMessage introduces support for phone numbers alongside email addresses, just like on iOS 6.

You can find the full changelog after the break. We’ll update this post with direct links as soon as they’re available.+

Update — You’ll find the direct downloads below:

• Client Update (665 MB)
• Combo Update (665 MB)

Facebook

• Single sign on for Facebook
• Adds Facebook as an option when sharing links and photos
• See Facebook friends’ contact information and profile pictures in Contacts
• Facebook notifications now appear in Notification Center

Game Center

• Share scores to Facebook, Twitter, Mail, or Messages
• Facebook friends are included in Game Center friend recommendations
• Added Facebook “Like” button for games
• Challenge friends to beat your score or achievement

Other new features

• Adds Power Nap support for MacBook Air (Late 2010)
• iMessages sent to your phone number now appear in Messages on your Mac*
• You can now add passes to Passbook (on your iPhone or iPod touch) from Safari and Mail on your Mac*
• FaceTime can now receive calls sent to your phone number*
• New shared Reminders lists
• New sort options allow you to sort notes by title, the date you edited them, and when you created them
• Dictation now supports additional languages: Mandarin, Cantonese, Spanish, Korean, Canadian English, Canadian French, and Italian
• Dictionary app now includes a French definition dictionary
• Sina Weibo profile photos can now be added to Contacts

General fixes

• Adds an option to discard the changes in the original document when choosing Save As
• Unsent drafts are now opened automatically when launching Mail
• Receive Twitter notifications for mentions and replies from anyone
• URLs are shortened when sending tweets from Notification Center
• Notifications are disabled when AirPlay Mirroring is being used
• Adds SSL support for Google searches from the Smart Search Field in Safari
• Adds a new preference to have Safari launch with previously open webpages
• Resolves an issue that may cause the “Enable Autodiscover” checkbox to always remain checked
• Enables access to the Mac App Store when Parental Controls are enabled
• Support for @icloud.com email addresses
• Resolves a video issue with some VGA projectors when connected to certain Mac notebooks
• Addresses an issue that may prevent Active Directory accounts from being locked out
• Resolves an issue that may cause the policy banner to re-appear prior to logging in
• Improvements to SMB
• Addresses an issue with NIS users when auto-login is enabled
• Addresses an issue in which the Keychain may not be accessible
• Ability to pre-authenticate a FileVault protected system
• Addresses an issue that may cause Xsan to not automatically start after migrating from Mac OS X Snow Leopard