Very surprising, but it is not possible to create group into the iPhone address book directly from the iPhone. The only way so far is to create them on your PC or Mac and synchronize it to the iPhone. After not possible to change a contact from one group to another or edit a group. Still you will need your Mac / PC address book to make the job. The only option on iPhone is to add a new user on an existing group.
On my side, I’m using Outlook. Into My Contacts, right click and select New Folder … to create a new group. After in iTunes the new group will appear and will be synchronize on your iPhone.