An unbroken string of 8 digits isn't terribly ergonomic...think of telephone numbers. Breaking things into chunks makes them both easier to remember and easier to communicate without errors. And as with phone numbers, consider a hierarchical system.
For example, use base 32, consisting of 10 decimal digits and 22 letters. A 3-symbol galactic sector code breaks the galaxy up into 32768 sectors. Another 3 symbols breaks each of those into 32768 sub-sectors, for just under 1 billion sub-sectors total. For local business, you only need the last three...there may be 32768 sub-sector A57's, but even near a border between sectors it will be pretty clear which one is meant, and you will rarely need to specify whether it's 3G9.A57 or 72E.A57.
Also, cylindrical coordinates centered on the core will result in inconveniently shaped sectors, only imperfectly fit the galaxy, and make little sense out in the halo, satellite galaxies, or other galaxies. Just put the galaxy in a Cartesian box. It is expandable: when you need to deal with locations outside a cube containing the Milky Way, just add a prefix defining intergalactic sectors, giving the one containing the Milky Way coordinates near the middle, so each Milky Way set of coordinates becomes GGG.###.###.
If you set the size of the Milky Way-containing intergalactic sector to 131072 light years across, then each galactic sector is 4096 light years across, and each subsector is 128 light years across. Or for a civilization with a bit less forethought, start defining sectors at the origin of the civilization and just add prefixes from there, with a result that doesn't so cleanly fit the Milky Way in a box.