#509097

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

Polycyclic presentation:

2^9|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, 416293);