Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C2×C5×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,13)(8,12,9,11)(14,15,18,21)(16,22,24,19)(17,20,26,28)(23,27,25,29);(6,8,10,9)(7,11,13,12)(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,27,28,29);(6,8,10,9)(7,12,13,11)(14,17,18,26)(15,20,21,28)(16,23)(19,27)(22,29)(24,25);(6,9,10,8)(7,12,13,11)(15,21)(19,22)(20,28)(27,29)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18937);