#170165

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

974 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,23)(18,21)(19,22)(20,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,37)(36,39)(38,40);(2,6)(3,4)(7,8)(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,20,22,23)(25,27,30,36)(26,31,34,40)(28,38,29,37)(32,39,33,35);(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)(2,6,8,7)(10,16)(14,21)(15,22)(20,23)(27,36)(31,40)(35,39)(37,38)

Polycyclic presentation:

2^9|b^2=h,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77361);