The chemical and structural properties of atomically precise nanoclusters are of great interest in numerous applications, but predicting the stable structures of clusters can be computationally expensive. In this work, we present a procedure for rapidly predicting low-energy structures of nanoclusters by combining a genetic algorithm with interatomic potentials actively learned on-the-fly. Applying this approach to aluminum clusters with 21 to 55 atoms, we have identified structures with lower energy than any reported in the literature for 25 out of the 35 sizes. Our benchmarks indicate that the active learning procedure accelerated the average search speed by more than an order of magnitude relative to genetic algorithm searches using only density functional calculations. This work demonstrates a feasible way to systematically discover stable structures for large nanoclusters and provides insights into the transferability of machine-learned interatomic potentials for nanoclusters.