#6513016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500019
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3173 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)(12,22,24,16)(13,18)(14,17)(19,28,23,31)(20,30)(21,29)(25,36,38,32)(26,34)(27,33)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,26)(14,21)(16,31)(17,29)(18,34)(19,24)(20,27)(22,28)(25,35)(30,33)(32,39)(36,40)(37,38);(2,5)(3,7)(9,12)(10,16)(11,19)(13,25)(14,24)(15,28)(17,22)(18,32)(20,35)(21,23)(26,37)(27,38)(29,31)(30,39)(33,36)(34,40);(1,3,4,7)(2,6,5,8)(10,17)(12,24)(15,29)(18,33)(19,23)(25,38)(30,34)(35,37);(2,5)(6,8)(9,13,14,27)(10,18,17,33)(11,20,21,26)(12,25,24,38)(15,30,29,34)(16,32,22,36)(19,35,23,37)(28,39,31,40)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,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, 6420212);