Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12805 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #410 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1799 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20)(10,23,24,13)(11,15)(12,17)(14,22)(18,32,33,28)(19,29)(21,27,35,34)(25,30,36,37)(26,31);(6,8,9,17,11,19,22,31)(7,12,14,16,15,26,20,29)(10,18,21,30,24,33,35,37)(13,25,27,28,23,36,34,32);(7,13)(12,25)(14,27)(15,23)(16,28)(20,34)(26,36)(29,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5793);