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) = C23 |
| direct product | C5×#11932 |
| 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 | |
398 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)(6,7,8,11,9,10,12,13)(14,15)(16,25)(17,22)(18,31)(19,35)(20,24)(21,27)(23,33)(26,39)(28,45)(29,42)(30,40)(32,38)(34,41)(36,43)(37,44);(7,10)(8,12)(14,16,20,29)(15,21,24,37)(17,28,19,26)(18,32,34,43)(22,38,35,36)(23,39,40,45)(25,41,42,31)(27,30,44,33);(14,17,20,19)(15,22,24,35)(16,26,29,28)(18,30,34,33)(21,36,37,38)(23,31,40,41)(25,39,42,45)(27,43,44,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4920);