C₂²×#57971 #72291785

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#57971

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

100477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 28 points:

(6,9)(7,11)(14,17)(15,19)(22,26)(24,28);(7,11)(10,12)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,28,27,24);(15,19)(18,20)(22,26)(23,27);(5,6,8,9)(7,10,11,12)(13,15,16,19)(14,18,17,20)(21,23)(22,24)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,27)(23,26)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8676694, C₂²×#8598543, C₂×#72231349, …
maximal normal subgroup of	C₂²×#8676694, C₂²×#8598543, C₂×#72231349, …
maximal quotient of	C₂²×#3860066, C₂²×#2108769, C₂²×#3332795, …

GAP

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

gap> G := Group((6,9)(7,11)(14,17)(15,19)(22,26)(24,28), (7,11)(10,12)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,28,27,24), (15,19)(18,20)(22,26)(23,27), (5,6,8,9)(7,10,11,12)(13,15,16,19)(14,18,17,20)(21,23)(22,24)(25,27)(26,28), (5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,27)(23,26)(25,28), (3,4), (1,2)(3,4));

C22×#57971 #72291785