#509421

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#33612	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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)(3,5,7,8)(9,10,14,17)(11,21,24,15)(12,22,27,40)(13,29,30,23)(16,19,33,20)(18,38,25,36)(26,37,35,28)(31,39,32,34);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,26)(13,29)(14,24)(16,25)(17,21)(18,33)(19,38)(20,36)(22,28)(23,30)(27,35)(31,34)(32,39)(37,40);(1,3,4,7)(2,5,6,8)(9,12,14,27)(10,13,17,30)(11,22,24,40)(15,34,21,39)(16,37,33,28)(18,35,25,26)(19,32,20,31)(23,38,29,36);(2,6)(5,8)(9,13,14,30)(10,12,17,27)(11,23,24,29)(15,35,21,26)(16,31,33,32)(18,34,25,39)(19,28,20,37)(22,38,40,36)

Polycyclic presentation:

2^9|a^2=e,c^2=e,d^2=i,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, 416617);