#9125947

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

4632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,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, 9033143);