C₂³×#35050 #72343065

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

Aliases

direct product

C₂³×#35050

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

3539832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

(3,4);(5,6);(21,25)(24,26);(8,10)(13,17)(15,18)(20,23)(21,25);(1,2)(3,4)(5,6);(13,17)(15,18);(11,12,14,16)(13,18,17,15)(19,20)(21,24)(22,23)(25,26);(7,8)(9,10)(11,13,14,17)(12,15,16,18)(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×#7642244
maximal normal subgroup of	C₂³×#7642244

GAP

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

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

C23×#35050 #72343065