#509101

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

678 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)(11,23)(12,27)(13,18)(14,19)(15,32)(16,22)(17,33)(20,25)(21,29)(24,38)(26,37)(28,36)(30,35)(31,34)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,18,19,23)(12,28,29,40)(15,25,30,24)(16,31,17,26)(20,32,38,35)(21,36,27,39)(22,37,33,34);(9,12)(10,19)(11,24)(13,25)(14,29)(15,16)(17,30)(18,34)(20,38)(21,35)(23,37)(26,40)(27,32)(28,31);(2,6)(3,8)(9,13)(10,20)(11,14)(12,25)(15,28)(16,31)(17,26)(18,35)(19,38)(21,34)(22,39)(23,32)(24,29)(27,37)(30,40)(33,36)

Polycyclic presentation:

2^9|b^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, 416297);