The rules are simple:
1) Chat must have been silent for a period of at least 10 minutes.
2) Screencaps must be posted, featuring the period of silence and the claim.
3) You must make the post in the chat, in the form of "/me claims the chat for <group>" (or something similar)
EDIT: For clarity, activity by yourself and the Forum Announcer does not count as breaking the 10 minutes of silence.
So, to get started:

The chat now belongs to the People!
-Gravis