C₂³×#2426 #72339192

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂³×#2426

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

148437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#59030, C₂³×#44927, C₂³×#36258, …
maximal normal subgroup of	C₂³×#59030, C₂³×#44927, C₂³×#36258, …

GAP

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

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

C23×#2426 #72339192