#9681730

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

4867 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^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,d^c=dh,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, 9588926);