#509290

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 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)(3,4)(5,7)(6,8)(9,10)(11,17)(12,14)(13,18)(15,20)(16,21)(19,27)(22,23)(24,36)(25,35)(26,37)(28,31)(29,30)(32,34)(33,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,16,26)(10,17,21,37)(12,23,28,40)(13,24,15,32)(14,22,31,39)(18,36,20,34)(19,25,30,38)(27,35,29,33);(1,4,5,6)(2,3,7,8)(10,18)(11,22)(13,15)(14,19)(17,33)(20,21)(23,36)(24,25)(26,39)(27,29)(30,31)(32,38)(34,40)(35,37);(2,7)(3,8)(9,12)(10,19)(11,23)(13,29)(14,18)(15,27)(16,28)(17,25)(20,31)(21,30)(22,36)(24,33)(26,40)(32,35)(34,39)(37,38)

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416486);