#7173219

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(25,26,29,31)(27,33,36,40)(28,37,34,32)(30,38,39,35);(1,3,4,7)(2,5,6,8)(11,17)(12,20)(14,21)(15,19)(25,27,34,38,29,36,28,35)(26,30,37,40,31,39,32,33);(2,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26)(27,30)(28,32)(29,31)(33,35)(34,37)(36,39)(38,40);(11,17)(14,21)(18,23)(22,24)(27,35)(28,34)(30,33)(32,37)(36,38)(39,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,31)(30,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,e^2=h,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, 7080415);