Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#7731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,17,19,18,16,20,21)(22,23)(24,31,33,36)(25,34)(26,29)(27,30)(28,32,37,35);(6,8,9,13)(7,10,11,12)(14,16)(15,18)(17,19)(20,21)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(7,11)(10,12)(15,16)(19,21)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 719);