#8807311

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

4791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714507);