#9014315

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8921511);