#6512999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499997
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	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3037 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:

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

Polycyclic presentation:

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

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, 6420195);