#9206658

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

5822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113854);