Armando Dippet is the Headmaster at this time. Albus Dumbledore is the Transfiguration teacher for several decades, starting in the 1930s. He eventually becomes the Headmaster in the mid-1960s.
We don't know exactly when Dumbledore took the Transfiguration post (except that it was before 1938), but we do know that he taught Transfiguration while Tom Marvolo Riddle was a student (HBP13, HBP20). He taught for over thirty years before taking over as Headmaster in the mid-1960s.