C₂³×#35076 #72339257

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

Aliases

direct product

C₂³×#35076

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

1929496 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7652795
maximal normal subgroup of	C₂³×#7652795

GAP

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

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

C23×#35076 #72339257