#8152206

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

4763 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)(17,18,24,22)(19,28,30,25)(20,26,21,31)(23,32,29,27);(1,2,6,5)(3,8,4,7)(11,15)(14,16)(17,19)(18,25)(20,23)(21,29)(22,28)(24,30)(26,27)(31,32);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,22,31)(19,23,30,29)(25,27,28,32);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,15)(13,16)(17,21,24,20)(18,26,22,31)(19,29,30,23)(25,27,28,32);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,22,24,18)(19,28,30,25)(20,31,21,26)(23,32,29,27);(17,23,24,29)(18,27,22,32)(19,20,30,21)(25,26,28,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8059402);