#9730344

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

4791 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)(3,6)(4,7)(5,8)(10,14)(11,16)(17,18)(19,22)(20,21)(23,24)(25,29)(26,32)(27,30)(28,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,26)(18,22,24,32)(20,27,29,31)(21,30,25,28);(9,11,13,16)(10,15,14,12)(18,24)(19,26)(20,29)(28,30);(9,12)(10,16)(11,14)(13,15)(17,20)(18,25)(19,27)(21,24)(22,28)(23,29)(26,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,21,23,25)(18,20,24,29)(19,28,26,30)(22,31,32,27);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,26)(19,24)(20,28)(21,27)(23,32)(25,31)(29,30)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,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, 9637540);