Social networks may contain privacy-sensitive information about individuals. The objective of the network anonymization problem is to alter a given social network dataset such that the number of anonymous nodes in the social graph is maximized. Here, a node is anonymous if it does not have a unique surrounding network structure. At the same time, the aim is to ensure data utility, i.e., preserve topological network properties and retain good performance on downstream network analysis tasks. We propose two versions of a genetic algorithm tailored to this problem: one generic GA and a uniqueness-aware GA (UGA). The latter aims to target edges more effectively during mutation by avoiding edges connected to already anonymous nodes. After hyperparameter tuning, we compare the two GAs against two existing baseline algorithms on several real-world network datasets. Results show that the proposed genetic algorithms manage to anonymize on average 14 times more nodes than the best baseline algorithm. Additionally, data utility experiments demonstrate how the UGA requires fewer edge deletions, and how our GAs and the baselines retain performance on downstream tasks equally well. Overall, our results suggest that genetic algorithms are a promising approach for finding solutions to the network anonymization problem.