#497702

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404898);