#6499999

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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3150 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)(11,15)(12,22,23,34)(13,18)(14,17)(16,24,31,26)(19,35,36,39)(20,30)(21,29)(25,33)(27,32)(28,37,40,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,29)(12,19,23,36)(13,25,27,20)(16,28,31,40)(18,33,32,30)(22,39,34,35)(24,38,26,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,31)(18,34)(20,38)(21,36)(22,32)(24,27)(25,37)(29,40)(30,39)(33,35);(3,8)(5,7);(2,6)(5,7)(10,17)(15,29)(16,31)(18,32)(22,34)(28,40)(30,33)(35,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 6407195);