Graph mining for structural patterns is a fundamental task in many applications. Compilation-based graph mining systems, represented by AutoMine, generate specialized algorithms for the provided patterns and substantially outperform other systems. However, the generated code causes substantial computation redundancy and the compilation process incurs too much overhead to be used online, both due to the inherent symmetry in the structural patterns. In this paper, we propose an optimizing compiler, GraphZero, to completely address these limitations through symmetry breaking based on group theory. GraphZero implements three novel techniques. First, its schedule explorer efficiently prunes the schedule space without missing any high-performance schedule. Second, it automatically generates and enforces a set of restrictions to eliminate computation redundancy. Third, it generalizes orientation, a surprisingly effective optimization that was mainly used for clique patterns, to apply to arbitrary patterns. Evaluated on multiple graph mining applications and complex patterns with 7 real-world graph datasets, GraphZero demonstrates up to 40X performance improvement and up to 197X reduction on schedule generation overhead over AutoMine.