Do you know a place where the integral cohomology of $G\_2$-homogeneous spaces is computed?


Great computational efforts using representation theory in order to determine the characteristic classes of homogeneous spaces were done by Borel and Hirzebruch in a series of papers: 


1. Characteristic classes and homogeneous spaces. I.
Amer. J. Math. 80 1958, 491–504.
2. Characteristic classes and homogeneous spaces. II.
Amer. J. Math. 81 1959 315–382.
3. Characteristic classes and homogeneous spaces. III.
Amer. J. Math. 82 1960 458–538.


I was unable to find there a systematic answer to the question. 



Reference requests: Integral cohomology of $G_2$-homogeneous spaces
I don't know how "systematic" the answer you are looking has to be, but for the quotients of the form $G\_2/T$ and $G\_2/P$ (P: Parabolic subgroup) you can find the results in [Schubert presentation of the integral cohomology ring of the flag manifolds G/T, by Haibao Duan and Xuezhi Zhao](https://arxiv.org/pdf/0801.2444.pdf)


