#9730316

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

4838 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)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26);(9,11,13,16)(10,15,14,12)(18,24)(19,27)(20,30)(26,31);(9,12)(10,16)(11,14)(13,15)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,21)(18,19)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,f^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, 9637512);