#6500009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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	#62875	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3129 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407205);