C₂³×#39862 #72325410

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵×C₄

Aliases

direct product

C₂³×#39862

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

1357352 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7837029
maximal normal subgroup of	C₂³×#7837029

GAP

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

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

C23×#39862 #72325410