#9027263

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

5139 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934459);