C₂²×#45563 #72326185

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#45563

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

85593 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)(21,22)(23,25)(24,27)(26,28);(6,9)(10,12)(22,23)(26,27);(14,17)(15,19);(3,4);(5,6)(7,12)(8,9)(10,11)(13,14,16,17)(15,18,19,20)(21,23,25,22)(24,27,28,26);(1,2)(3,4);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598997
maximal normal subgroup of	C₂²×#8598997
maximal quotient of	C₂²×#873665

GAP

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

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

C22×#45563 #72326185