#8943238

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

4404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,e^2=h,f^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, 8850434);