We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group G2. For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler 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 G2-invariant almost complex structures on these manifolds.