#8726240

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

4773 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=i,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, 8633436);