C₂²×#57189 #72301284

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

Aliases

direct product

C₂²×#57189

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

77849 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)(7,8)(9,11)(10,12)(21,22,24,27)(23,25,28,26);(5,7,9,12)(6,10,11,8)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,28)(26,27);(7,12)(8,10)(13,15)(14,17)(16,20)(18,19)(21,22,24,27)(23,25,28,26);(5,8)(6,7)(9,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,12)(9,11)(13,14)(15,17)(16,19)(18,20)(21,23)(22,26)(24,28)(25,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8757652
maximal normal subgroup of	C₂²×#8757652

GAP

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

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

C22×#57189 #72301284