#8607433

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

4625 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26)(27,34)(28,31)(29,36)(30,32)(33,38)(35,39)(37,40);(2,5)(6,7)(10,15)(11,19)(12,22)(21,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,18)(20,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,23,17,21)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=g,f^2=h,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, 8514629);