There are multiple assignment types that can be used in Canvas.

Regular Canvas assignments, course Group assignments, Peer Review, Turnitin (full Turnitin and Plagiarism Framework), and VoiceThread

Note - grade center categories are also called groups in canvas, so if you're looking to create a Group Assignment you need to look for "Course Group" not assignment group.

Please see the links below for more information on creation, grading, etc. of assignments:

Assignments https://community.canvaslms.com/docs/DOC-4131#jive_content_id_Assignments

Turnitin assignments https://mizzou.freshdesk.com/solution/categories/1000127493

VoiceThread assignments https://mizzou.freshdesk.com/solution/folders/1000212826