C₂³×#39844 #72325206

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

Aliases

direct product

C₂³×#39844

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

1311688 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)(11,12)(13,18)(14,16)(15,17)(21,25)(24,26);(8,10)(11,13,15,14)(12,16,17,18)(19,20,23,22)(21,26,25,24);(3,4);(5,6);(7,8)(9,10)(19,21)(20,24)(22,26)(23,25);(1,2)(3,4)(5,6);(11,12)(13,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25);(11,14,15,13)(12,16,17,18)(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×#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((8,10)(11,12)(13,18)(14,16)(15,17)(21,25)(24,26), (8,10)(11,13,15,14)(12,16,17,18)(19,20,23,22)(21,26,25,24), (3,4), (5,6), (7,8)(9,10)(19,21)(20,24)(22,26)(23,25), (1,2)(3,4)(5,6), (11,12)(13,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25), (11,14,15,13)(12,16,17,18)(19,22,23,20)(21,24,25,26));

C23×#39844 #72325206