My coop and run all together is 30Lx30Wx6H with 2 mini coop housing units on pallets inside it that are 3Lx4Wx4H, and I am putting shade netting on the top to not only keep them in, but to keep them cool. I have 45 birds all together, so hopefully that will give them enough room. Right now they are in a 40x40 with my goats, and it seems too big at times. I do know that more room seems to cut down on plucking and general smacking up of the smaller ones.
If your coop is 6x6x6 that sounds like cubic feet to me. What you should do is multiply length x width to get square feet. So that's 6'x6' which is 36 square feet.
Now from what I remember you should look at 4 sq foot as a minimum, so you are looking at 9 birds.