#509102

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

682 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)(11,18)(12,26)(13,30)(14,32)(15,22)(16,33)(17,19)(20,23)(21,29)(24,37)(25,34)(27,38)(28,39)(31,36)(35,40);(1,3)(2,5)(4,8)(6,7)(9,11,17,13)(10,18,19,30)(12,27,29,40)(14,24,28,23)(15,31,16,25)(20,32,37,39)(21,35,26,38)(22,36,33,34);(9,12)(10,19)(11,23)(13,24)(14,15)(16,28)(17,29)(18,34)(20,37)(21,39)(25,40)(26,32)(27,31)(30,36);(2,6)(3,8)(9,13,17,11)(10,20,19,37)(12,24,29,23)(14,27,28,40)(15,31,16,25)(18,32,30,39)(21,34,26,36)(22,38,33,35)

Polycyclic presentation:

2^9|b^2=i,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, 416298);