#6451754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3841 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)(3,8)(4,6)(5,7)(9,10)(11,20)(12,17)(13,18)(14,26)(15,19)(16,22)(21,36)(23,31)(24,33)(25,35)(27,29)(28,34)(30,37)(32,38)(39,40);(9,11,15,23)(10,16,19,29)(12,21,25,39)(13,22,14,27)(17,32,35,30)(18,31,26,20)(24,37,28,38)(33,36,34,40);(1,3,5,4)(2,6,7,8)(13,14)(16,29)(18,26)(22,27)(24,28)(30,32)(33,34)(37,38);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,24)(14,28)(15,25)(16,30)(18,33)(19,35)(20,36)(22,37)(23,39)(26,34)(27,38)(29,32)(31,40);(9,13,15,14)(10,18,19,26)(11,22,23,27)(12,24,25,28)(16,31,29,20)(17,33,35,34)(21,37,39,38)(30,40,32,36)

Polycyclic presentation:

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

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, 6358950);