#6454812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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	#62810	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) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362008);