The more mass the greater the gravity. Mass equals gravity.
Since gravity is produced by mass, the center of mass is also the center of gravity. The only difference between these two concepts is that mass is a more basic quantity, so the center of mass would also be the center of inertia, as well as the center of gravity. In practice, these terms can be used interchangeably.
For a uniform symmetric body in all directions the center of mass and center of gravity are the same point.
The center of mass and center of gravity serve two separate purposes. As an individual, your center of gravity and center of mass remains the same. The earth has a center of mass. The moon has a center of mass. The moon does not orbit the earth. They orbit their center of gravity. Their center if gravity is always changing. It is not a fixed point. When an airplane takes off, its center of gravity is one place. As it burns fuel, its center of gravity changes. It might be necessary to change its attitude, which today is done automatically by computers. The computers are checked at the end of the flight by human beings.
That all depends on the shape of the object and how its mass is distributed. The center of gravity of a solid sphere is at the center of the solid sphere. The center of gravity of a solid cube is at the center of the solid cube. The Earth's center of gravity is at the center of the Earth, and there's certainly plenty of mass there. But the center of gravity of a ring is at the center of the ring ... an open space where the finger goes.
The center of Gravity can be defined as the point where the weight of the all body depends is called Center of Gravity or Center of mass.
Toward the center of mass of the object or objects attracting you. Gravity also pulls it/them toward the center of mass of you.
If the object is homogeneous, its center of mass is in its geometrical center. And if it is small compared to Earth, its center of gravity is, for all practical purposes, its center of mass.
Center of gravity = center of mass, for most practical purposes. Mainly, the center of mass is the place where you imagine the mass to be concentrated. However, as a result, you can also consider inertia and weight to be concentrated at the center of mass.