#9009230

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
prime radical	#57270	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8916426);