#9222160

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

4986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9129356);