#8720637

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

4457 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)(3,7)(4,5)(6,8)(10,14)(11,17)(16,19)(18,20)(25,26,31,35)(27,36,38,32)(28,33,37,29)(30,34,39,40);(2,5)(6,7)(9,10,13,14)(11,15,17,22)(12,16,21,19)(18,23,20,24)(25,27,31,38)(26,32,35,36)(28,30,37,39)(29,34,33,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,22)(12,16,21,19)(18,23,20,24)(25,28,31,37)(26,33,35,29)(27,30,38,39)(32,40,36,34);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,28)(26,29)(27,30)(31,37)(32,34)(33,35)(36,40)(38,39);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,29)(26,28)(27,34)(30,32)(31,33)(35,37)(36,39)(38,40);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,30)(26,34)(27,37)(28,38)(29,36)(31,39)(32,33)(35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627833);