#509696

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#25434	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^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, 416892);