Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12814 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(14,15)(16,24)(17,20)(18,30)(19,33)(21,26)(22,32)(23,29)(25,40)(27,44)(28,41)(31,39)(34,38)(35,42)(36,43)(37,45);(6,7,8,9)(15,20)(21,35)(22,36)(23,33)(24,40)(30,41)(37,39)(38,44);(14,16,17,25)(15,21,20,35)(18,31,28,45)(19,34,29,27)(22,38,36,44)(23,39,33,37)(24,41,40,30)(26,43,42,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5802);