#168539

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

855 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)(3,5)(4,7)(6,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(28,35)(30,36)(31,38)(33,39);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)(25,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,38)(39,40);(2,5)(3,8)(6,7)(25,27)(26,30)(28,33)(29,34)(31,37)(32,36)(35,39)(38,40);(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

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

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, 75735);