#168439

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^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, 75635);