Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#18715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3549 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,19,24)(16,21,26,29)(17,23,18,22)(20,27,28,25);(8,12)(10,13)(14,16)(15,20)(17,25)(18,27)(19,26)(21,22)(23,29)(24,28);(6,7)(8,13)(9,11)(10,12)(14,16)(15,21)(17,27)(18,25)(19,26)(20,22)(23,28)(24,29);(14,17,19,18)(15,22,24,23)(16,25,26,27)(20,21,28,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11703);