#7173218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080414);