C₂³×#2311 #72324526

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂³×#2311

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

333653 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#35512, C₂³×#34256, C₂³×#35250, …
maximal normal subgroup of	C₂³×#35512, C₂³×#34256, C₂³×#35250, …
maximal quotient of	C₂³×#13956, C₂³×#13957, C₂³×#14345, …

GAP

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

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

C23×#2311 #72324526