#8389273

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

4452 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)(3,6)(4,5)(7,8)(9,10,15,18)(11,16,19,14)(12,13,17,24)(20,21,22,23)(25,26)(27,36)(28,30)(29,39)(31,35)(32,38)(33,37)(34,40);(1,2,4,5)(3,6,7,8)(9,11,15,19)(10,16,18,14)(12,22,17,20)(13,23,24,21)(25,27)(26,32)(28,29)(30,34)(31,38)(33,39)(35,36)(37,40);(3,7)(6,8)(11,19)(14,16)(20,22)(21,23)(25,28,31,37)(26,33,35,30)(27,29,38,40)(32,39,36,34);(2,5)(6,8)(9,12)(10,13)(11,20)(14,23)(15,17)(16,21)(18,24)(19,22)(25,27)(26,32)(28,29)(30,34)(31,38)(33,39)(35,36)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,21)(12,18)(14,20)(15,24)(16,22)(19,23)(25,29)(26,34)(27,37)(28,38)(30,36)(31,40)(32,33)(35,39);(1,3,4,7)(2,6,5,8)(9,14)(10,11)(12,23)(13,20)(15,16)(17,21)(18,19)(22,24)(25,30,31,33)(26,28,35,37)(27,34,38,39)(29,36,40,32)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,f^2=g,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, 8296469);