#335571

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10046	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=eh,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242767);