#8429368

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

4893 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)(3,7)(10,13)(14,16)(17,18,23,28)(19,22,27,24)(20,26,21,25)(29,32,30,31);(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,19,23,27)(18,24,28,22)(20,29,21,30)(25,32,26,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,25,28,26)(19,29,27,30)(22,31,24,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,20)(18,26,28,25)(19,30,27,29)(22,32,24,31);(17,22,23,24)(18,27,28,19)(20,31,21,32)(25,30,26,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336564);