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) = C4 |
| direct product | C5×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
262 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)(8,11)(9,10)(12,13)(14,15,20,25,18,23,34,26)(16,24,29,38,28,21,19,35)(17,30,32,42,31,45,33,37)(22,39,41,44,40,36,43,27);(7,10)(11,13)(14,16,20,29,18,28,34,19)(15,21,25,35,23,24,26,38)(17,27,32,39,31,44,33,36)(22,37,41,30,40,42,43,45);(6,8,9,12)(7,11,10,13)(14,17,18,31)(15,22,23,40)(16,27,28,44)(19,36,29,39)(20,32,34,33)(21,37,24,42)(25,41,26,43)(30,38,45,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6101);