#6454857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62851	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362053);