#8689245

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

4425 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8596441);