Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21851 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,6,10,24,35,7,11,25,36,26,34,50,58)(3,13,16,37,17,39,45,59,15,8,22,29,43,28,48,54)(4,18,20,38,21,31,47,56,19,9,23,32,46,33,49,57)(14,30,41,55,42,51,61,63,40,27,44,52,60,53,62,64);(1,3)(2,8)(4,14)(5,22)(6,17)(7,15)(9,30)(10,28)(11,13)(12,37)(16,25)(18,27)(19,40)(20,44)(21,42)(23,41)(24,48)(26,43)(29,36)(31,53)(32,52)(33,51)(34,39)(35,59)(38,55)(45,50)(46,60)(47,62)(49,61)(54,58)(56,63)(57,64);(1,4)(2,9)(3,14)(5,23)(6,21)(7,19)(8,27)(10,33)(11,18)(12,38)(13,30)(15,40)(16,44)(17,42)(20,25)(22,41)(24,49)(26,46)(28,53)(29,55)(31,34)(32,36)(35,56)(37,52)(39,51)(43,60)(45,62)(47,50)(48,61)(54,63)(57,58)(59,64);(2,10,11,34)(4,19)(8,28,13,39)(9,31,18,33)(12,35,36,58)(14,40)(20,23)(21,46)(27,51,30,53)(29,54,37,59)(32,56,38,57)(41,44)(42,60)(47,49)(52,63,55,64)(61,62)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=ce,f^b=fi,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403668);