C₂⁴×#2431 #72291296

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

Aliases

direct product

C₂⁴×#2431

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

6259112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#35680
maximal normal subgroup of	C₂⁴×#35680

GAP

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

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

C24×#2431 #72291296