#9434201

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

4703 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)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26);(2,5)(3,7)(9,10,12,14)(11,13,16,15)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32);(1,3)(2,6)(4,7)(5,8)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26);(1,2)(3,6)(4,5)(7,8)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341397);