#509777

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

689 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)(3,7)(4,6)(5,8)(9,10,15,20,16,21,13,14)(11,17,30,35,24,33,23,19)(12,25,28,18,29,31,27,34)(22,36,39,32,38,26,37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,26)(13,23)(14,19)(15,30)(16,24)(18,22)(20,35)(21,33)(25,37)(27,32)(28,40)(29,36)(31,39)(34,38);(9,12)(10,18)(11,22)(13,28)(14,31)(15,27)(16,29)(17,32)(19,26)(20,25)(21,34)(23,39)(24,38)(30,37)(33,40)(35,36);(2,6)(3,8)(9,11)(10,19)(12,22)(13,30)(14,17)(15,23)(16,24)(18,26)(20,33)(21,35)(25,40)(27,39)(28,37)(29,38)(31,32)(34,36)

Polycyclic presentation:

2^9|a^2=fi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416973);