Multicast group is a group of hosts that are communicating with each other using multicast. IGMP is the protocol that makes this grouping possible.
You dont assign class D IPs, however, your computer might be using one as it is a member of that particular group. If you have two mulicast applications running on your network, they will have different class D address.
A very simple example is OSPF. You dont assign your OSPF router address 224.0.0.5. But still it (application on your router) uses it and communicates with other OSPF peers via this address.
Hope this helps.