C₂³×#35840 #72325216

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

Aliases

direct product

C₂³×#35840

Subgroups

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

1360376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7918389
maximal normal subgroup of	C₂³×#7918389

GAP

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

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

C23×#35840 #72325216