How Big was Rome's "Pentagon"?

At its peak, the (Western) Roman Army had over 300,000 soldiers. Presumably, the Romans had some sort of central command to handle purchasing (of armor, swords, uniforms, etc.), and educate the officer classes…I would assume they had to have some kind of finance dept., to handle paying the troops.
So, did they have something like the Pentagon? how big was it? And, was it a plum position 9as a Pentagon posting is today)?

For starters, a lot of the logistics were handled locally, as were payroll details. The level of centralization you’re accustomed to was impossible when the fastest speed at which documents could travel was that of a horse its rider didn’t care to keep alive…

Theres some information in this article. Details seem a bit sketchy. I’m not sure the scholars understand how the daily command structure worked.

That’s not quite true. There was a vast amount of central planning, especially after the reforms of Diocletian. A central budget was computed, and the military upkeep expense was allocated to the provinces proportionally at a very fine level of detail. The last thing the state wanted was to let local officials control military compensation. Most of the grain for soldier pay was grown in Egypt, anyway. So it all had to be expropriated, shipped up the Nile, and distributed all over the empire.

The central officials in charge of military supply were the praetorian prefects. They were typically only just a smidgen less powerful than the emperors themselves. Even though they eventually had no military power, their authority was nearly absolute and their competence was vast. The prefecture did have a finance department that became responsible for the military budget.

There was no Pentagon per se because the administration was frequently on the road.

By the late empire, about half of the war matieriel was constructed at state operated manufactories called fabricae. The rest was made up from private purchases. Other military equipment, like clothing, was either purchased or taxed directly. There was a brief cloak tax documented in Egypt, where small groups of people were collectively responsible for producing cloaks for military issue.