#6452259

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	#62856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3231 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,17)(13,19)(14,18)(15,20)(16,25)(22,31)(23,36)(24,32)(26,39)(27,35)(28,34)(29,33)(30,38)(37,40);(9,11,14,25,15,22,13,24)(10,16,19,21,20,32,18,31)(12,26,27,38,28,23,29,37)(17,30,33,39,34,40,35,36);(1,3,5,4)(2,6,7,8)(11,22)(13,14)(18,19)(21,31)(23,26)(27,29)(33,35)(36,39);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,30)(18,33)(19,35)(20,34)(21,36)(22,26)(24,37)(25,38)(31,39)(32,40);(9,13,15,14)(10,18,20,19)(11,24,22,25)(12,27,28,29)(16,31,32,21)(17,33,34,35)(23,37,26,38)(30,39,40,36)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6359455);