To delete a Group you must be the manager of that Group, or be a community administrator. You can check who is the manager of a Group by going to the Profile View of the Group and clicking 'Edit' on the 'More options' menu.
Deleting a Group
If you are the manager of a Group you can delete it by navigating to either the Feed View or Profile View of the Group and clicking the 'Delete' button on the right hand side.
Please be aware that once the group is deleted any discussions, tasks or documents that were shared to that group will still exist within BCSocial, but access rights between some users and the items may be lost if the Group in question was providing the necessary permissions.
To learn more about permissions please view this article.