#8188566

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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)(3,8,4,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,31,33)(27,32,29,39)(28,36,38,30)(34,35,37,40);(1,3,6,4)(2,7,5,8)(10,14)(15,17)(16,21)(22,23)(25,27,31,29)(26,32,33,39)(28,37,38,34)(30,35,36,40);(2,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,38)(26,30,33,36)(27,34,29,37)(32,40,39,35);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(1,4,6,3)(2,8,5,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,27)(26,32,33,39)(28,37,38,34)(30,40,36,35);(1,5,6,2)(3,8,4,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30)(26,28)(27,35)(29,40)(31,36)(32,37)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095762);