#6584339

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60457	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 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)(3,8)(4,6)(5,7)(9,10,14,12)(11,20,23,17)(13,18,29,25)(15,26,16,19)(21,24,34,31)(22,37,38,33)(27,40,39,35)(28,30,36,32);(1,3,5,4)(2,6,7,8)(9,11,14,23)(10,17,12,20)(13,27,29,39)(15,24,16,31)(18,35,25,40)(19,34,26,21)(22,30,38,32)(28,33,36,37);(9,12,15,26,14,10,16,19)(11,21,24,17,23,34,31,20)(13,25,30,36,29,18,32,28)(22,35,27,33,38,40,39,37);(1,4,5,3)(2,6,7,8)(9,12,14,10)(11,20,23,17)(13,28,29,36)(15,26,16,19)(18,30,25,32)(21,31,34,24)(22,35,38,40)(27,33,39,37);(9,13)(10,18)(11,22)(12,25)(14,29)(15,30)(16,32)(17,33)(19,28)(20,37)(21,35)(23,38)(24,27)(26,36)(31,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491535);