#336773

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 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)(3,5,7,8)(9,10,12,17)(11,18,19,14)(13,15,21,16)(20,23,24,22)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(26,31)(27,36)(28,30)(32,39)(35,40)(37,38);(1,3,4,7)(2,5,6,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(2,6)(5,8)(10,16)(11,19)(13,21)(14,22)(15,17)(18,23)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243969);