[snip of some interesting galactic engineering]
It will take a while to accomplish all this. At the speed of light,
your acquisition probes will take about 50Kyears to reach the limits
of a 50K lightyear sphere surroundng the galaxy. Once there, they can
start sending all the matter back to the collection point. Assuming
transit speeds of .5c, you will expend two thirds of the mass to
accelerate the other third at the start and decelerrate it at the
end of the trip, so after an additional 100K years, you will have
accumulated "only" one third of the galaxy's mass. If you are
willing to wait, you can use longer transit times and end up with
more mass. Of course, you will be able to begin construction with
material accumulated much earlier in the program, as soon
as you reach the galactic center, about 25Kyears from now.