#336165

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,17)(13,15)(14,19)(16,21)(20,23)(22,24)(26,31)(32,39)(33,34)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,26,29,31)(27,35,37,40)(28,34,30,33)(32,38,39,36);(1,4,5,6)(2,3,7,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,35)(34,40);(2,7)(3,8)(10,16)(11,19)(13,21)(14,22)(15,17)(18,23)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,35,39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ehi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243361);