#72267751

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72227399
commutator, hypocenter	G	C₁
Frattini	C₂⁴×C₄	#11757163
Fitting, cosocle	C₂⁸×C₄	A₅
socle	C₂⁵	#14515303
LCS	G
UCS	C₁ ◁ C₂

Only the 9 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 9 normal subgroups are shown; non-normal subgroups are omitted.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 58 points:

(2,5,3)(4,8,7)(11,14,50,15,25,52,26,32,39,20,37,27)(12,57,18,19,45,33,36,30,13,21,51,22)(16,43,48,29,53,35,38,58,17,34,55,31)(23,47,49,28,54,42,46,56,24,41,44,40);(1,4,6,3)(5,9,7,10)(11,14,15,25,26,32,20,37)(12,18,19,33,36,13,21,22)(16,28,34,23,38,41,29,46)(17,24,31,40,48,49,35,42)(27,44,50,56,52,54,39,47)(30,43,51,55,57,58,45,53)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,5,3)(4,8,7)(11,14,50,15,25,52,26,32,39,20,37,27)(12,57,18,19,45,33,36,30,13,21,51,22)(16,43,48,29,53,35,38,58,17,34,55,31)(23,47,49,28,54,42,46,56,24,41,44,40), (1,4,6,3)(5,9,7,10)(11,14,15,25,26,32,20,37)(12,18,19,33,36,13,21,22)(16,28,34,23,38,41,29,46)(17,24,31,40,48,49,35,42)(27,44,50,56,52,54,39,47)(30,43,51,55,57,58,45,53));