Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#8409 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,9,11,10,13,8,12)(14,15,18,23,19,22,33,37)(16,24,27,38,26,20,31,35)(17,28,30,39,29,21,32,36)(25,40,42,45,41,34,43,44);(7,11)(8,9)(12,13)(14,16)(15,20)(17,25)(18,31)(19,26)(21,34)(22,24)(23,38)(27,33)(28,40)(29,41)(30,43)(32,42)(35,37)(36,45)(39,44);(14,17)(15,21)(16,25)(18,32)(19,29)(20,34)(22,28)(23,39)(24,40)(26,41)(27,43)(30,33)(31,42)(35,45)(36,37)(38,44)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1397);