Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#25270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,21,17,22,18,23)(16,24,27,28,25,20,26,29);(7,10)(11,13)(14,16,18,26,17,25,19,27)(15,20,23,28,22,24,21,29);(6,8,9,12)(7,11,10,13)(15,21)(16,25)(18,19)(20,28)(22,23)(24,29);(7,10)(11,13)(15,22)(20,24)(21,23)(28,29)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18258);