Gestapo Command Headquarters was a military-political complex located on Ekos in 2268. As part of that planet's Nazi government, the Gestapo Command Headquarters served (as the name implied) as headquarters for the Ekosian Gestapo.
When two phasers were confiscated from members of an Enterprise landing party, the weapons (designated as "alien pistols") were sent to the Gestapo Command Headquarters for analysis. The laboratory facilities were apparently more advanced than those to be found in the Ekosian Chancellery. (TOS: "Patterns of Force")