#509103

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

665 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)(11,15)(12,23)(13,30)(14,19)(16,21)(17,25)(18,24)(20,36)(22,34)(26,40)(27,29)(28,31)(32,38)(33,35)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,17,19,15)(12,27,28,39)(13,30,18,24)(16,26,34,20)(21,36,22,40)(23,37,31,29)(32,35,33,38);(9,12,14,28)(10,18,19,13)(11,23,25,31)(15,35,17,38)(16,37,34,29)(20,39,26,27)(21,32,22,33)(24,40,30,36);(2,6)(3,8)(9,13)(10,12)(11,24)(14,18)(15,27)(16,32)(17,39)(19,28)(20,35)(21,29)(22,37)(23,40)(25,30)(26,38)(31,36)(33,34)

Polycyclic presentation:

2^9|b^2=i,c^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, 416299);