#6516747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3314 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)(11,19,22,33)(12,16)(13,18)(14,17)(15,21,29,25)(20,36,38,35)(23,31)(24,30)(26,34)(27,32)(28,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,29)(18,33)(19,32)(21,27)(23,39)(24,38)(26,37)(30,40)(31,35)(34,36);(2,5)(3,7)(9,12,14,24)(10,16,17,30)(11,20,22,38)(13,26,27,23)(15,28,29,40)(18,34,32,31)(19,35,33,36)(21,39,25,37);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,30)(18,32)(19,33)(28,40)(31,34)(35,36);(2,5)(6,8)

Polycyclic presentation:

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

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, 6423943);