C₂²×#4033889 #72297012

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄²

Aliases

direct product

C₂²×#4033889

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

749720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#72231349
maximal normal subgroup of	C₂²×#72231349

GAP

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

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

C22×#4033889 #72297012