Find the location of its center, using a compass only.

The following procedure is sufficient to locate the centre of the circle, which is what is asked for.

Set the point of the compass at any point, X, on the edge of the original circle C1, and draw another smaller circle, X1. Repeat the exercise with the point of the compass at one intersection of the two circles, and the pencil end starting at X, the centre of X1.

By construction, these two circles are of equal size and their two points of intersection DE are on a line that is a diameter of C1.

Repeat the same exercise with two more smaller circles on some other point on C1, to generate points of intersection  F and G. Lines DE and FG are both diameters of C1, so they intersect at its centre.

Constructing circles that pass through the centre of C1 is also possible but is a bit more complicated.