C₂²×#4122655 #72296054

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#4122655

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

591896 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231761
maximal normal subgroup of	C₂²×#72231761

GAP

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

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

C22×#4122655 #72296054