C₂²×#48473 #72296214

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

Aliases

direct product

C₂²×#48473

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

79245 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811
maximal normal subgroup of	C₂²×#9228811
maximal quotient of	C₂²×#2171267

GAP

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

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

C22×#48473 #72296214