Deep squats
A study published in the Journal of Strength & Conditioning Research found that compared to shallow squats and parallel squats, deep squats engage the gluteus maximus better, especially in the concentric phase. (3)
This is logical, since the lower you go, the more you stretch (and thereby include in the work) the gluteal muscles. Shallow squats engage the thigh muscles more than the glutes. Therefore, go lower for the full work of the gluteus maximus muscle.
How to do this exercise:
Place the barbell on your shoulders (top of the trapezoid), look straight, straighten your chest, legs slightly wider than shoulder-width, toes turned slightly to the sides.
Tighten your abdominal muscles and move your pelvis back and down (as if sitting on a chair), focus on your heels, and keep your back straight throughout the exercise. Rise, pushing off the heels from the floor and squeezing the buttocks.