Re: Punching power and buttock size/strength. How important are the glutes?
Glutes are hip extensors- They are a part of a bigger picture of muscles required for punching.
Any time that the femur ends up behind the body, the hip extensors are making that happen. Include the hamstrings if you want to include he other part of hip extensors.
There are tons of factors and variables you can throw in there when talking about what muscles are involved with a specific punch.