#6445422

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

2989 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,23)(21,36)(22,29)(24,35)(25,32)(26,34)(28,33)(30,39)(31,37)(38,40);(9,11,15,24)(10,16,19,29)(12,21,26,38)(13,23,14,22)(17,31,34,30)(18,35,27,20)(25,39,28,37)(32,36,33,40);(1,3,4,8)(2,5,6,7)(13,14)(16,29)(18,27)(22,23)(25,28)(30,31)(32,33)(37,39);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,30)(18,32)(19,34)(20,36)(22,37)(23,39)(24,38)(27,33)(29,31)(35,40);(9,13,15,14)(10,18,19,27)(11,22,24,23)(12,25,26,28)(16,20,29,35)(17,32,34,33)(21,37,38,39)(30,36,31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^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, 6352618);