#9040980

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

4812 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)(11,14)(12,13)(15,16)(17,18,23,25)(19,27,29,24)(20,30,32,21)(22,31,28,26);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,23,29)(18,24,25,27)(20,31,32,26)(21,28,30,22);(2,5)(3,7)(10,12)(11,15)(18,25)(21,30)(22,28)(24,27);(2,5)(6,8)(11,15)(14,16)(17,20,23,32)(18,21,25,30)(19,26,29,31)(22,27,28,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,20)(19,28)(22,29)(23,30)(24,31)(25,32)(26,27);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,22,23,28)(18,26,25,31)(19,21,29,30)(20,27,32,24)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948176);