#8899361

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

4473 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)(11,16)(12,14)(13,18)(15,17)(19,21)(20,22)(23,24)(25,26)(27,36)(28,39)(29,40)(30,34)(31,32)(33,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,20)(10,16,17,22)(12,18,19,24)(13,21,23,14)(25,27,32,38)(26,33,31,36)(28,37,30,29)(34,35,39,40);(2,5)(6,8)(10,17)(11,20)(14,21)(18,24)(25,28,32,30)(26,34,31,39)(27,37,38,29)(33,35,36,40);(3,7)(6,8)(9,12,15,19)(10,14,17,21)(11,18,20,24)(13,22,23,16)(25,29)(26,35)(27,28)(30,38)(31,40)(32,37)(33,39)(34,36);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,14)(12,16)(15,23)(17,24)(19,22)(20,21)(25,30,32,28)(26,34,31,39)(27,37,38,29)(33,40,36,35);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,13)(12,17)(15,21)(16,24)(18,22)(20,23)(25,31,32,26)(27,36,38,33)(28,34,30,39)(29,40,37,35)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=hi,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, 8806557);