#8720563

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

4624 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=i,f^2=g,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, 8627759);