...I came up with all those government departments...a-and it's vitally important to know the colour of officials' robes is based on their level...
And lucky for you, I have an idea of how you can still do it.
Alice: Why'd they have to choose such an ugly color for the robes worn by ____?
Bob: It's because the color ____ was associated with ____ by the ____s due to ____.
Alice: Well what about the ____'s robes?
Bob: They're colored ____ because...
And there you have it. Just fill in the blanks and continue on, and you'll have yourself some exposition, along with characterizing Bob as being knowledgeable about history or whatever.
Also, there are people who don't pay that much attention to how their government is run. People usually call them Americans. Self deprecating humor
aside, this type of person can provide you an opportunity to drop some exposition. Have a dumber character (or just one that doesn't care how the government works as long as it does) complaining about what the leader is doing, only for the character that knows how the government works to correct him/her on various things (mostly which department is responsible for which problems).