#509250

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33522	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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416446);