A galactic core is the center of a galaxy. Also referred to as a galactic bulge or galactic center.
They are areas dense with stars, usually in the rotational center of a galaxy(where some surmise supermassive blackholes exist, around which the rest of the galaxy spins). Like, earth is on one of the arms of our Milky Way galaxy, and the closest star is over 4 lightyears away. One parsec is about 3.26 lightyears across.
Within one parsec of our galactic core, there are as many as 10 million stars. So, much more dense with stars, but still spread out over multiple lightyears of distance. Again, a lightyear is the space light covers in a whole year. Our sun is 8 light
minutes away. Again for reference of scale, a year has 525,600 seconds.
So no, a galactic core is not just one solar system. It is tens of millions of solar systems, crammed into several lightyears of space*which is still huge.