#501851

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33526	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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)(9,10)(11,18)(12,26)(13,17)(14,20)(15,32)(16,21)(19,29)(22,36)(23,34)(24,33)(25,38)(27,39)(28,40)(30,35)(31,37);(1,3,4,7)(2,5,6,8)(10,17)(11,22)(13,29)(14,15)(18,26)(19,20)(21,35)(23,30)(24,27)(25,31)(32,34)(33,36)(37,39)(38,40);(2,6)(5,8)(9,11,16,25)(10,18,21,38)(12,23,28,29)(13,24,30,27)(14,31,15,22)(17,33,35,39)(19,26,34,40)(20,37,32,36);(9,12,16,28)(10,18,21,38)(11,23,25,29)(13,22,30,31)(14,27,15,24)(17,26,35,40)(19,33,34,39)(20,36,32,37)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,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, 409047);