#8899044

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

4506 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26)(27,36)(28,39)(29,40)(30,34)(31,32)(33,38)(35,37);(1,2)(3,6)(4,5)(7,8)(11,12)(15,18)(19,20)(23,24)(25,27,32,38)(26,33,31,36)(28,37,30,29)(34,35,39,40);(2,5)(6,8)(9,11,14,12)(10,15,17,18)(13,20,22,19)(16,24,21,23)(25,28,32,30)(26,34,31,39)(27,37,38,29)(33,35,36,40);(3,7)(6,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,29)(26,35)(27,28)(30,38)(31,40)(32,37)(33,39)(34,36);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,30,32,28)(26,34,31,39)(27,37,38,29)(33,40,36,35);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,31,32,26)(27,36,38,33)(28,34,30,39)(29,40,37,35)

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806240);