#6599128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506324);