Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
2084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,18)(16,23,20,21)(17,25,27,29)(22,28,26,24);(7,10)(11,13)(14,16,19,20)(15,21,18,23)(17,24,27,28)(22,25,26,29);(14,17)(15,22)(16,24)(18,26)(19,27)(20,28)(21,25)(23,29);(6,8,9,12)(7,11,10,13)(14,18,19,15)(16,23,20,21)(17,26,27,22)(24,29,28,25);(14,16,19,20)(15,23,18,21)(17,24,27,28)(22,29,26,25)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53575);