#9070702

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

5049 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8977898);