#8935040

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842236);