C₂³×#2440 #72339283

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

Aliases

direct product

C₂³×#2440

Subgroups

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

144661 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#36123, C₂³×#36215, C₂³×#59026, …
maximal normal subgroup of	C₂³×#36123, C₂³×#36215, C₂³×#59026, …

GAP

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

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

C23×#2440 #72339283