In order to block a member from a group, you must be a Group Owner or Admin for the group.
Here's how to block a member from a group:
1. While in the group select the "Member List" under the 'More Admin Tools' link.
2. Enter the member's screen name.
3. Click the "remove from group" button.
A pop up will appear asking for you to just "Remove" the member or "Remove and Block".