#9224235

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

4757 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,13,15,16)(18,20)(19,27)(22,32)(25,29);(1,3,5,7)(2,4,6,8)(11,15)(13,16)(17,18,23,20)(19,28,27,24)(21,29,31,25)(22,26,32,30);(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,28)(21,22)(23,27)(25,26)(29,30)(31,32);(2,6)(4,8)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26);(1,2)(3,4)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,22)(20,29)(23,31)(24,26)(27,32)(28,30);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,26)(19,21)(20,30)(23,32)(24,25)(27,31)(28,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=i,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, 9131431);