Engrima18/cayley-graphs-513-to-576

Cayley Graphs — orders 513–576

This dataset contains Cayley graphs of finite groups: one row per group, covering groups whose order is between 513 and 576. Each graph is the Cayley graph built from the group's minimal generating set (see Provenance below).

• Rows (groups): 9,782
• Group orders covered: 513–576 (64 distinct orders)
• Task: binary graph classification (default label: IsMonolithic).

About the CayleyNet collection

This dataset is part of a census of 131,406 Cayley graphs covering every finite group of order at most 767 (except order 512), built to study how finite-group structure is reflected in the network geometry of Cayley graphs. Each group is recorded with exact algebraic property labels alongside a broad collection of graph, cycle, distance, and spectral statistics. The census provides benchmarks for predicting group properties directly from graph data — comparing classical models, an MLP, and graph neural networks (GIN/GCN) — and contributes new OEIS sequences for monolithic groups and for groups generated by at most 3, 4, and 5 elements.

Code: https://github.com/Engrima18/CayleyNet

Columns

Column	Type	Description
GroupOrder	int32	Order of the finite group (first entry of the GAP SmallGroup id).
GroupIndex	int32	Index of the group among all groups of that order (second entry of the GAP SmallGroup id).
AdjMatrixNonZerEnt	large_string	Directed edge list of the Cayley graph as a JSON-style nested list [[src, dst], ...]; nodes are 0-indexed group elements.
EdgeFeatures	large_string	One-hot generator matrix of shape [num_edges, num_generators]; row e indicates which generator produced edge e.
MinNumOfGens	int16	Size of a minimal generating set of the group.
IsAbelian	bool	Whether the group is abelian.
IsNilpotent	bool	Whether the group is nilpotent.
IsSimple	bool	Whether the group is simple.
IsPerfect	bool	Whether the group is perfect (G = [G, G]).
IsSolvable	bool	Whether the group is solvable.
IsMonolithic	bool	Whether the group is monolithic, i.e. has a unique minimal normal subgroup. Primary classification label.
IsCyclic	bool	Whether the group is cyclic.

Group-property / label balance

Column	# True	% True
IsAbelian	134	1.37%
IsNilpotent	732	7.48%
IsSimple	8	0.08%
IsPerfect	0	0.00%
IsSolvable	9,780	99.98%
IsMonolithic	29	0.30%
IsCyclic	64	0.65%

Numeric column statistics

Column	Min	Mean	Max	Nulls
GroupOrder	513	572.4	576	0
GroupIndex	1	3,860	8,681	0
MinNumOfGens	1	3.148	6	0
NumEdges	521	4,414	4,608	0

Parsing the list-valued columns

AdjMatrixNonZerEnt and EdgeFeatures are stored as strings holding a JSON-style nested list. Decode them with:

import ast
edges = ast.literal_eval(row["AdjMatrixNonZerEnt"])   # [[src, dst], ...]
edge_feats = ast.literal_eval(row["EdgeFeatures"])    # one-hot [E, n_gens]

Provenance

• Generated with GAP / SageMath and NetworkX (scripts/generate_data.py).
• Distributed as typed Parquet: the source Id is split into GroupOrder and GroupIndex, and integer statistics are stored as nullable integers.

Usage

from datasets import load_dataset

ds = load_dataset("Enrico18/cayley-graphs-513-to-576", split="train")
print(ds[0])

License

MIT — © 2025 Enrico Grimaldi.