#8636311

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

4849 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gi,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, 8543507);