Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60618 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C23×D4 |
| prime radical | C2×C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C42×C5 ◁ G | |
2170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,22)(17,21)(18,23)(19,24)(20,25)(26,29)(27,28);(7,10)(11,13)(14,16,19,27,20,26,18,17)(15,21,23,29,25,28,24,22);(6,8,9,12)(7,11,10,13)(14,17,18,26,20,27,19,16)(15,22,24,28,25,29,23,21);(16,26)(17,27)(21,28)(22,29);(14,18,20,19)(15,23,25,24)(16,17,26,27)(21,29,28,22)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53606);