#167531

order	512
nilpotency class	3
isoclinism	isoclinic class of #166993
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

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, 74727);