#9164027

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

5100 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071223);