Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3486 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,14,25,29)(8,16,19,30,18,12,23,27)(9,20,21,31,22,13,24,28)(17,26,34,36,33,32,35,37);(6,8)(7,12)(9,17)(10,23)(11,18)(13,26)(14,16)(15,30)(19,25)(20,32)(21,35)(22,33)(24,34)(27,29)(28,37)(31,36);(12,16)(26,32)(27,30)(36,37);(6,9,11,22)(7,13,14,20)(8,17,18,33)(10,24,25,21)(12,26,16,32)(15,31,29,28)(19,35,23,34)(27,37,30,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14768);