More than one band has used this name:
1. The Breakfast Club was formed in New York in the late 1970s, and went through several line-ups including one in which Madonna was the drummer. By the early 1980s, the band included Madonna, Angie Smit on bass, and the Gilroy brothers, Dan and Ed, both on guitar (Dan sang lead vocals as well). Dan Gilroy was also briefly Madonna's boyfriend, and he eventually allowed her to sing some lead vocals. Madonna ultimately left to form a new band Emmy, but the remaining members soldiered on. Read more on Last.fm