C₂²×#56344 #72293727

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

Aliases

direct product

C₂²×#56344

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

90429 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:

(5,6,9,11)(7,8,12,10)(13,14)(15,16)(17,20)(18,19)(21,22)(23,28)(24,26)(25,27);(5,7)(6,10)(8,11)(9,12)(13,15,17,19)(14,18,20,16)(21,23,25,26)(22,24,27,28);(5,7,9,12)(6,8,11,10)(13,16,17,18)(14,19,20,15)(21,24)(22,26)(23,27)(25,28);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28);(5,8)(6,7)(9,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,26)(23,27)(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×#8598543, C₂²×#8898065, C₂×#72231820
maximal normal subgroup of	C₂²×#8598543, C₂²×#8898065, C₂×#72231820
maximal quotient of	C₂²×#4029469, C₂²×#2171267

GAP

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

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

C22×#56344 #72293727