#8665381

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572577);