Interconnection network plays an essential role in the architecture of large-scale high performance computing (HPC) systems. In the paper, we construct a novel family of low-diameter topologies, Galaxyfly, using techniques of algebraic graphs over finite fields. Galaxyfly is guaranteed to retain a small constant diameter while achieving a flexible tradeoff between network scale and bisection bandwidth. Galaxyfly lowers the demands for high radix of network routers and is able to utilize routers with merely moderate radix to build exascale interconnection networks. We present effective congestion-aware routing algorithms for Galaxyfly by exploring its algebraic property. We conduct extensive simulations and analysis to evaluate the performance, cost and power consumption of Galaxyfly against state-of-the-art topologies. The results show that our design achieves better performance than most existing topologies under various routing algorithms and traffic patterns, and is cost-effective to deploy for exascale HPC systems.