#8633970

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

4719 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8541166);