#8402268

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

4605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 8309464);