C₂⁴×#2688 #72325781

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

Aliases

direct product

C₂⁴×#2688

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

2135416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 8 generators on 28 points:

(9,10)(11,12)(13,14,16,18)(15,17,20,19)(21,22)(23,27)(24,25)(26,28);(9,10)(11,12)(13,15,16,20)(14,17,18,19)(21,23)(22,26)(24,27)(25,28);(3,4);(5,6);(10,12)(13,15)(14,17)(16,20)(18,19)(21,24,25,22)(23,27,28,26);(7,8);(9,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28);(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×#1420724
maximal normal subgroup of	C₂³×#1420724

GAP

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

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

C24×#2688 #72325781