#9039214

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946410);