#6575439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2103 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)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,31)(24,32)(25,33)(26,35)(27,36)(28,34)(29,38)(30,37)(39,40);(2,5)(6,8)(9,11,16,25)(10,17,22,33)(12,14,24,15)(13,26,29,39)(18,32,21,19)(20,35,38,40)(23,30,27,28)(31,37,36,34);(10,18)(11,12)(14,15)(19,32)(20,34)(21,22)(23,26)(24,25)(27,39)(28,30)(35,40)(37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,14)(13,27)(15,25)(16,24)(17,18)(20,36)(21,33)(22,32)(23,29)(26,28)(30,39)(31,38)(34,35)(37,40);(9,13,16,29)(10,20,22,38)(11,23,25,27)(12,26,24,39)(14,28,15,30)(17,31,33,36)(18,34,21,37)(19,35,32,40)

Polycyclic presentation:

2^9|b^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,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, 6482635);