When we walk, we instinctively swing our right shoulder forward as the left leg/hip steps forward; and reversing this, the left shoulder advances as the right leg/hip steps forward. This is called "opposition" and we walk with a natural, organic oppositional motion of the shoulders and hips.
This oppositional, rotating "core body action" of the torso is the way we moved as infants crawling on the floor. To experience this clearly, anyone should just try crawling on all fours for a bit.
When we stand and walk a few steps, "core power" feels like a "twist at the waist" initiated from the body's core torso muscles - not created peripherally by just swinging the arms and legs.
Even though we swim in an elongated (and flattened) body shape, we achieve "power from the core" by rotating the trunk of the body as if we were walking.
In today's class and video, you'll see the student excellently demonstrating this bilateral movement.