#8213868

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

4627 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8121064);