The Executive Council (執行部, Shikōbu) is the main force among the students of Todo Academy, becoming 3rd year enforcement group. They keep the order on the school and punish the students that do not follow their rules or in a response to string of violent behavior student could exhibit.
Originally Maya remembers the "Punishment" started as a means for the student council to control public morals. She thought it had became just a formality at school, and at worst the brats will just be beaten to a point where they will be able to recover (it's false due Tsutomu Ryuuzaki raped Chiaki in Chapter #5 to give a punishment Bob and Nagi).
The Executive Council is chosen annually by the Election Tournament, where the winning group becomes the new Executive Council for the next school term. Two years ago, Mitsuomi became the leader of the group and his group won the next, keeping their post.