#8107620

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

4757 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:

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

Polycyclic presentation:

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

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, 8014816);