#9143409

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

5039 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050605);