#6584351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
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	#60460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(9,10,13,17)(11,20,23,15)(12,16,28,35)(14,19,31,18)(21,24,33,25)(22,37,39,32)(26,38,40,34)(27,29,36,30);(1,3,5,4)(2,6,7,8)(9,11,13,23)(10,15,17,20)(12,26,28,40)(14,24,31,25)(16,34,35,38)(18,33,19,21)(22,30,39,29)(27,37,36,32);(9,10,14,18,13,17,31,19)(11,21,24,15,23,33,25,20)(12,16,29,27,28,35,30,36)(22,38,40,32,39,34,26,37);(1,4,5,3)(2,6,7,8)(9,10,13,17)(11,20,23,15)(12,27,28,36)(14,18,31,19)(16,29,35,30)(21,25,33,24)(22,34,39,38)(26,32,40,37);(9,12)(10,16)(11,22)(13,28)(14,29)(15,32)(17,35)(18,27)(19,36)(20,37)(21,38)(23,39)(24,40)(25,26)(30,31)(33,34)

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,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6491547);