#9789120

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

4985 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,f^2=gh,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, 9696316);