C₂⁴×#2669 #72325784

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷×C₄

Aliases

direct product

C₂⁴×#2669

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5023432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1259281
maximal normal subgroup of	C₂³×#1259281

GAP

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

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

C24×#2669 #72325784