#369166

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(5,6,8,7)(9,10)(11,21)(12,26)(13,30)(14,31)(15,20)(16,19)(17,24)(18,28)(22,34)(23,37)(25,36)(27,40)(29,39)(32,35)(33,38);(5,7,8,6)(9,11,16,32,15,25,14,24)(10,17,31,36,20,35,19,21)(12,27,29,22,28,38,13,23)(18,34,39,40,26,37,30,33);(1,2,3,4)(9,12,14,13,15,28,16,29)(10,18,31,39,20,26,19,30)(11,22,32,38,25,23,24,27)(17,33,21,37,35,40,36,34);(9,13,14,28,15,29,16,12)(10,18,19,30,20,26,31,39)(11,23,24,38,25,22,32,27)(17,34,21,33,35,37,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276362);