Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12916 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1850 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,11,17,12,16,31,42,13,18,32,43,10,27,30,44)(8,19,23,46,21,14,33,36,22,37,52,62,28,35,55,61)(9,15,25,39,24,40,53,64,26,41,54,65,29,38,56,63)(20,47,50,68,51,59,57,69,49,45,66,67,48,34,58,60);(6,8,11,23,12,21,31,33,13,22,32,52,10,28,30,55)(7,14,17,36,16,37,42,62,18,35,43,61,27,19,44,46)(9,20,25,50,24,51,53,57,26,49,54,66,29,48,56,58)(15,34,39,60,40,47,64,68,41,59,65,69,38,45,63,67);(6,9)(7,15)(8,20)(10,29)(11,25)(12,24)(13,26)(14,34)(16,40)(17,39)(18,41)(19,45)(21,51)(22,49)(23,50)(27,38)(28,48)(30,56)(31,53)(32,54)(33,57)(35,59)(36,60)(37,47)(42,64)(43,65)(44,63)(46,67)(52,66)(55,58)(61,69)(62,68)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5904);