#8824163

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

4884 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)(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,23)(24,27)(26,29)(28,32)(30,31);(2,5)(7,8)(9,10,13,16)(11,14,15,12)(18,23)(21,24)(25,31)(29,32);(1,2,6,5)(3,8,4,7)(9,10,13,16)(11,12,15,14)(17,19,22,27)(18,21,23,24)(20,28,30,26)(25,32,31,29);(1,3,6,4)(2,7,5,8)(9,11,13,15)(10,14,16,12)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28);(1,5,6,2)(3,8,4,7)(9,12,13,14)(10,15,16,11)(17,21,22,24)(18,19,23,27)(20,29,30,32)(25,26,31,28)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=hi,e^2=i,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, 8731359);