#9787151

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

4750 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gh,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, 9694347);