#168467

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75663);