#9788909

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

4691 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696105);