C₂²×#7699549 #72301074

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7699549

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1609960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232033
maximal normal subgroup of	C₂²×#72232033

GAP

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

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

C22×#7699549 #72301074