Groups are the primary entities that make up a crowd. Understanding group-level dynamics and properties is thus scientifically important and practically useful in a wide range of applications, especially for crowd understanding. In this paper, we show that fundamental group-level properties, such as intra-group stability and inter-group conflict, can be systematically quantified by visual descriptors. This is made possible through learning a novel collective transition prior, which leads to a robust approach for group segregation in public spaces. From the former, we further devise a rich set of group-property visual descriptors. These descriptors are scene-independent and can be effectively applied to public scenes with a variety of crowd densities and distributions. Extensive experiments on hundreds of public scene video clips demonstrate that such property descriptors are complementary to each other, scene-independent, and they convey critical information on physical states of a crowd. The proposed group-level descriptors show promising results and potentials in multiple applications, including crowd dynamic monitoring, crowd video classification, and crowd video retrieval.