#8650370

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

4910 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557566);