#8726710

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

4868 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633906);