In humans, the gluteus maximus muscle is an extensor of the pelvis, it does not play an important role in walking, however, with increasing load, such as fast walking or running, the muscles of the buttocks come into action to vigorously extend the pelvis and straighten the trunk.
Exercises for the buttocks

These biomechanical observations make it clear that when doing exercises such as “good morning” and lifting the barbell from the floor with straight legs using heavy weight, the gluteus maximus muscles are more involved and the hamstring muscles are much less involved.
The gluteal muscles, like the deltoids, consist of three parts and are called the gluteal delta.

Lunges in the Smith machine.
This exercise primarily loads the gluteus maximus muscle. Smith machine lunges are one of the best exercises for beautiful buttocks.

Shoulder lunges
This exercise primarily targets the gluteus maximus.

Lunges with dumbbells
This exercise mainly targets the gluteus maximus and quadriceps muscles.

Back leg swing from the bottom block
This exercise primarily engages the gluteus maximus and, to a lesser extent, the tibialis ischio, excluding the short head of the biceps femoris.

Leg swings back with the lever of the simulator
This exercise primarily targets the gluteus maximus, and to a lesser extent the semitendinosus, semimembranosus, and long head of the biceps femoris.

Back leg swing on the floor
This exercise, performed with a straight leg, develops the hamstrings and the gluteus maximus. With the leg bent at the knee, only the gluteus maximus muscle is developed, but less intensively.

“Bridge” lying
This exercise primarily targets the hamstrings and gluteus maximus muscles.