Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,18,24,19,25,33,39,20,22,34,40,32,41,17,23)(16,26,28,36,29,42,27,43,30,44,31,21,35,37,45,38);(6,7,9,11)(8,13,12,10)(14,16)(15,21)(17,31)(18,27)(19,35)(20,30)(22,36)(23,42)(24,26)(25,43)(28,33)(29,32)(34,45)(37,39)(38,41)(40,44);(15,22)(21,36)(23,39)(24,40)(25,41)(26,44)(37,42)(38,43)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4923);