In order for a group member to create contacts, you must assign the basic right, Create Contact records. In addition, this right requires the assignment of View Contact records.
The Create Contact records right requires the assignment of the View Contact records right. As a result, a group member with Create Contact records right will have access to all pages that result from the View Contact record right. In addition, the Create Contact record right will result in a New button that is available on the Contact List page.
Contacts List Page (New Button)