#6445420

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,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, 6352616);