We consider the class A_Γ of n-dimensional normed spaces with unit balls of the form: B_U = conv ∪ γ∈Γ γ(B_n ¹∪U (B_n ¹)), where B_n ¹ is the unit ball of ℓ_n ¹, Γ is a finite group of orthogonal operators acting in ℝⁿ, and U is a "random" orthogonal transformation. It is proved that this class contains spaces with a large Banach-Mazur distance between them. If the cardinality of Γ is of order n^c, it is shown that, in the power scale, the diameter of A_Γ in the modified Banach-Mazur distance behaves as the classical diameter and is of order n. Bibliography: 8 titles.