Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
Determining whether one finite group is isomorphic to a subgroup of another is a fundamental problem in computational group theory. In this work, we propose a Siamese Graph Neural Network (Siamese GNN) for subgroup prediction using Cayley graph representations of finite groups. Each input group is represented by its undirected Cayley graph and encoded by one branch of a Siamese GNN to produce a graph embedding. The resulting graph embeddings are combined with algebraic features derived directly from the input groups to construct a joint feature vector, which is processed by a fully connected classifier to predict subgroup relations between finite groups. By integrating graph-based structural representations with algebraic features, the proposed framework provides a unified approach for learning subgroup relations from finite groups. Experimental results demonstrate the effectiveness of the proposed architecture, achieving a test accuracy of 95.9% (47/49) on an independent test set and illustrating the potential of geometric deep learning for subgroup prediction.
In this work, we present a general Graph Neural Network (GNN) framework for learning algebraic properties of finite groups from their Cayley graph representations. The framework provides a unified computational pipeline consisting of a common graph construction procedure, feature representation, training methodology, and GNN architecture, with only the target labeling function varying across classification tasks. To demonstrate the generality of the proposed approach, we consider three representative algebraic properties: abelianity, nilpotency, and solvability. Experiments were conducted on a benchmark of 176 finite groups drawn from several classical families, with all groups included in each classification task. To address class imbalance, class-weighted cross-entropy loss was employed where appropriate during training. Furthermore, the family PSL(2,q) was reserved exclusively for testing, enabling evaluation of the framework's ability to generalize to previously unseen group families. The best-performing models achieved test balanced accuracies of 1.000, 0.856, and 0.875 for abelianity, nilpotency, and solvability, respectively. Although the same computational framework was employed across all tasks, different GNN architectures proved optimal for different algebraic properties, suggesting that the representational complexity required to learn a property depends on its underlying algebraic structure. These results demonstrate that GNNs can effectively learn multiple algebraic properties directly from Cayley graph representations while exhibiting strong generalization to unseen group families. More broadly, the proposed framework establishes a computational methodology for studying algebraic properties of finite groups using graph-based machine learning.