Engrima18/cayley-stats-256

Cayley Group Statistics — order 256

This dataset contains graph-theoretic statistics of finite groups: one row per group, covering groups of order 256. Each graph is the Cayley graph built from the group's minimal generating set (see Provenance below).

• Rows (groups): 56,092
• Group orders covered: 256–256 (1 distinct order)
• 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).
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.
AvgClusteringCoeff	float64	Average clustering coefficient of the graph.
Density	float64	Edge density of the graph.
DiameterApprox	int16	Approximate graph diameter.
DiameterDistMeas	int16	Graph diameter computed via exact distances.
Girth	int16	Girth (length of the shortest cycle); 0 if acyclic.
GutmanIndex	int64	Gutman index of the (undirected) graph.
NuOfTrianglesOutOfEachNode	int32	Number of triangles incident to a node (equal for all nodes by vertex-transitivity of Cayley graphs).
SchultzIndex	int64	Schultz index of the (undirected) graph.
SquareClusteringCoeff	float64	Square clustering coefficient.
WeinerIndex	int64	Wiener index (sum of shortest-path distances).
CycleLengthsAndFreq	large_string	numpy-repr string [array(lengths) array(freqs)]: for the minimum cycle basis, the distinct cycle lengths and their frequencies.

Group-property / label balance

Column	# True	% True
IsAbelian	22	0.04%
IsNilpotent	56,092	100.00%
IsSimple	0	0.00%
IsPerfect	0	0.00%
IsSolvable	56,092	100.00%
IsMonolithic	1,130	2.01%
IsCyclic	1	0.00%

Numeric column statistics

Column	Min	Mean	Max	Nulls
GroupOrder	256	256	256	0
GroupIndex	1	2.805e+04	56,092	0
MinNumOfGens	1	4.398	8	0
AvgClusteringCoeff	0	0.07113	0	0
Density	0	0.03137	0	0
DiameterApprox	4	5.242	8	0
DiameterDistMeas	4	5.242	8	0
Girth	3	3.02	4	0
GutmanIndex	6,881,280	1.402e+07	20,995,200	0
NuOfTrianglesOutOfEachNode	0	8.771	21	0
SchultzIndex	1,720,320	2.307e+06	2,910,208	0
SquareClusteringCoeff	0	0.02204	0	0
WeinerIndex	200,960	2.222e+05	262,144	0

Parsing CycleLengthsAndFreq

This column is a raw NumPy repr such as [array([3 4]) array([2 2])], meaning cycle lengths [3, 4] occur with frequencies [2, 2] in the minimum cycle basis. It is kept verbatim from the source.

Provenance

• Generated with GAP / SageMath and NetworkX (scripts/generate_data.py and scripts/process_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-stats-256", split="train")
print(ds[0])

License

MIT — © 2025 Enrico Grimaldi.