Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#7407 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #353 |
| commutator | C2×M16 | C2×C4×C5 |
| Frattini, cosocle | #408 | C22×C5 |
| socle | C2×C5 | #1343 |
| prime radical | C5×#2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C5×#408 ◁ G | |
240 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,10,15,12,16,29,44)(8,18,20,47,22,48,52,67)(9,14,25,38,26,40,58,42)(11,31,30,61,24,17,45,43)(13,33,34,55,35,62,63,23)(19,36,49,41,50,46,51,66)(21,54,53,69,32,37,28,60)(27,59,57,68,56,39,65,64);(6,8)(7,13)(9,23)(10,28)(11,32)(12,22)(14,37)(15,41)(16,35)(17,46)(18,39)(19,27)(20,45)(21,24)(25,51)(26,55)(29,53)(30,52)(31,36)(33,57)(34,43)(38,47)(40,54)(42,67)(44,66)(48,59)(49,58)(50,56)(60,68)(61,63)(62,65)(64,69)155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 395);