Many animals, including humans, have evolved to live and move in groups. In humans, disrupted social interactions are a fundamental feature of many psychiatric disorders.  However, we know little about how genes regulate social behavior.  Zebrafish may serve as a powerful model to explore this question.  By comparing the behavior of wild-type fish to 90 genetic lines, we show that mutations of genes associated with human psychiatric disorders can alter the collective behavior of adult zebrafish.  We identify three categories of behavioral variation across mutants: "scattered", in which fish show reduced cohesion; "coordinated", in which fish swim more in aligned schools; and "huddled", in which fish form dense but disordered groups. Changes in individual interaction rules can explain these differences.  This work demonstrates how emergent patterns in animal groups can be altered by genetic changes in individuals, and establishes a framework for understanding the fundamentals of social information processing.