#509100

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

Polycyclic presentation:

2^9|c^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, 416296);