#6501032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi

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, 6408228);