#170517

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
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

657 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^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,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77713);