#10086346

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

4703 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993542);