Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12678 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #442 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1776 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
292 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)(6,7,10,17,12,18,26,16,13,19,27,33,11,15,9,25)(8,20,22,31,23,32,36,35,24,34,37,14,28,30,21,29);(6,8,11,23,13,24,12,28)(7,14,15,29,19,31,18,35)(9,22,27,36,26,37,10,21)(16,32,17,34,25,30,33,20);(7,15,19,18)(9,26)(10,27)(14,29,31,35)(16,33,25,17)(20,30,34,32)(21,36)(22,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5666);