Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26633 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13310 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,15,34,51,9,14,36,50,8,16,35,52)(3,17,20,40,22,54,59,42,21,55,58,41,23,47,31,10)(4,24,25,48,37,39,62,60,19,11,32,44,26,53,57,64)(5,27,28,12,18,45,33,43,29,49,63,38,30,61,56,46);(1,3,9,21)(2,10,14,42)(4,5,19,29)(6,31,36,59)(7,23,8,22)(11,43,24,46)(12,39,38,53)(13,47,50,54)(15,41,16,40)(17,51,55,52)(18,37,30,26)(20,34,58,35)(25,56,32,33)(27,60,49,64)(28,62,63,57)(44,45,48,61);(1,4)(2,11)(3,18)(5,23)(6,32)(7,37)(8,26)(9,19)(10,38)(12,42)(13,48)(14,24)(15,53)(16,39)(17,45)(20,56)(21,30)(22,29)(25,36)(27,47)(28,59)(31,63)(33,58)(34,57)(35,62)(40,46)(41,43)(44,50)(49,54)(51,60)(52,64)(55,61);(1,5,8,18,9,29,7,30)(2,12,15,46,14,38,16,43)(3,19,22,37,21,4,23,26)(6,33,35,63,36,56,34,28)(10,39,41,11,42,53,40,24)(13,49,51,45,50,27,52,61)(17,44,47,64,55,48,54,60)(20,57,59,32,58,62,31,25)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=fh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411678);