#9210384

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

4666 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9117580);