Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13016 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1879 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
462 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)(6,7,12,18,13,19,35,51,14,20,11,36,34,50,10,17)(8,21,25,44,26,49,23,52,27,55,24,15,39,42,37,43)(9,28,30,53,31,64,29,16,32,46,33,47,40,45,38,48)(22,56,58,63,59,41,61,54,60,67,65,69,62,68,57,66);(6,8,13,39,14,27,34,26)(7,15,19,52,20,44,50,43)(9,22,40,59,32,60,31,62)(10,24,12,23,35,25,11,37)(16,41,53,67,48,68,47,56)(17,49,18,21,51,42,36,55)(28,63,45,54,46,69,64,66)(29,58,30,61,38,65,33,57);(6,9)(7,16)(8,22)(10,33)(11,38)(12,29)(13,40)(14,32)(15,41)(17,28)(18,45)(19,53)(20,48)(21,54)(23,58)(24,57)(25,61)(26,62)(27,60)(30,35)(31,34)(36,64)(37,65)(39,59)(42,69)(43,56)(44,68)(46,51)(47,50)(49,63)(52,67)(55,66)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6004);