This Apple doc should help ya out -> iOS: Syncing with Google Contacts
Follow these steps to set up contact syncing with Google Contacts:
- Tap Settings > Mail, Contacts, Calendars > Add Account > Other > Add CardDav Account.
- Enter the following information in the fields:
- Server: google.com
- User Name: Enter your full Google email address
- Password: Your Google account password
- Select Next at the top of the screen to complete the setup.
After you have completed the setup, open the Contacts app on your device. Syncing should begin automatically.
Note: Make sure that SSL is enabled (under Advanced settings), and that the port is 443.
From what I can tell, it looks like syncing Google contact groups via CardDAV isn't supported yet. Take a look at this Google Groups thread -> Does CardDAV Sync Support Groups?
I think ABC contacts will do that for you, but I haven't used it before. There is a FAQ here -> iPhone ABContacts Application