#8664800

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

4986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8571996);