Here's a breakdown:

* Local Group: This is a collection of about 50 galaxies, including our own. It's a relatively small grouping of galaxies compared to larger structures like galaxy clusters.

* Center of the Local Group: This is a region of space where the gravitational influence of all the galaxies within the Local Group is strongest. It's not a physical object, but rather a point in space where the gravitational pull is balanced.

So, while our galaxy is massive and has its own rotation, it's also moving through space, orbiting the center of the Local Group. This journey takes billions of years.

Important Note: The Local Group itself is also moving through space, likely being influenced by the gravity of even larger structures like galaxy clusters.