C₂²×#58995 #72278431

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

Aliases

direct product

C₂²×#58995

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

131069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7674648
maximal normal subgroup of	C₂²×#7674648

GAP

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

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

C22×#58995 #72278431