You have it reversed. The brownshirts were the SA, which were lead by a lot of homosexuals like Ernst Rohm. The blackshirts were the SS, which began as part of the SA, and led by Himmler. The Gestapo and the SS moved against the SA on the night you referred to.
Thanks for the correction. Did think that the Blackshirts were part of the SA but forgot about the later split.
It is like the Bolsheviks killing the Trotskyites. Mensheviks, etc. Last man standing concept - wins all the marbles and the power.