  We consider the proper class of all metric spaces endowed with the
Gromov--Hausdorff distance. Its maximal subclasses, consisting of the spaces on
finite distance from each other, we call clouds. Multiplying all distances in a
metric space by the same positive real number, we obtain a similarity
transformation of the Gromov--Hausdorff class. In our previous work, we
observed that with such a transformation, some clouds can jump to others. To
characterize the phenomenon, we studied the stabilizers of the similarity
action. In this paper, we prove that every cloud with a nontrivial stabilizer
has a center, i.e., a metric space for which all similarities from the
stabilizer generate a new space at zero distance. Moreover, the center is
unique modulo zero distance. The proof is based on the cloud completeness
theorem.
