#9050417

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

4522 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)(27,34)(28,32)(29,37)(30,31)(33,38)(35,36)(39,40);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27,30,32)(26,28,31,34)(29,35,38,39)(33,36,37,40);(1,3)(2,5)(4,7)(6,8)(10,14)(11,17)(12,20)(22,24)(26,31)(28,34)(29,38)(35,39);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,38)(26,33,31,37)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|b^2=hi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8957613);