#9053889

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

4232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(3,7)(5,8)(9,10,14,12)(11,15,18,17)(13,16,21,20)(19,22,24,23)(26,29)(27,33)(32,36)(35,39);(1,2)(3,5)(4,6)(7,8)(10,12)(11,18)(13,21)(22,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29,31,26)(27,34,33,30)(28,36,38,32)(35,40,39,37);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,34)(26,33,29,27)(28,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961085);