#507946

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,h^c=hi,f^d=fi

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, 415142);