#9786731

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

4799 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:

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

Polycyclic presentation:

2^9|b^2=ghi,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=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, 9693927);