#8639841

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8547037);