Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18656 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,19,23)(16,24,27,28)(17,20,18,22)(21,26,29,25);(6,7,10,13)(8,11,9,12)(15,20)(17,18)(21,28)(22,23)(24,29)(25,26);(6,8,10,9)(7,11,13,12)(14,16,19,27)(15,21,23,29)(17,25,18,26)(20,28,22,24);(6,8,10,9)(7,12,13,11)(14,17,19,18)(15,22,23,20)(16,25,27,26)(21,24,29,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11644);