#6619942

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	#39185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(19,26)(20,21)(23,31)(24,29)(27,32)(28,30);(2,6)(7,8)(10,14)(11,12)(17,18,22,25)(19,27,28,23)(20,29,21,24)(26,32,30,31);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,30)(22,28)(24,32)(25,27)(29,31);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,18)(19,27)(20,29)(21,24)(22,25)(23,28)(26,32)(30,31);(9,12,13,11)(10,16,14,15)(17,20,22,21)(18,24,25,29)(19,26,28,30)(23,32,27,31)

Polycyclic presentation:

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

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, 6527138);