We discuss the complex geometry of two complex five-dimensional K\"ahler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all K\"ahlerian complex structures was proved by Brieskorn, for the other one we prove it here... We relate the K\"ahler assumption in Brieskorn's theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all $G_2$-invariant almost complex structures on these manifolds. (read more)