#8533547

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

4511 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440743);