If you're talking about mail list groups, you can fool the Contacts app into doing this with a third party app called MailShot.
If you're talking about organization groups, you can only create these if you have an iCloud account, and they have to be created in the web app at icloud.com (using a desktop browser). Or you can do it in the Calendar app on a Mac, if you have one and sync it to the same iCloud account.