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×#60588 |
| 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 | |
2134 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)(16,25)(17,23)(18,24)(19,21)(20,28)(22,26)(27,29);(7,10)(11,13)(15,21)(22,29)(23,25)(24,28);(14,16,19,17)(15,22,21,29)(18,26,20,27)(23,28,25,24);(6,8,9,12)(7,11,10,13)(14,17,19,16)(15,23,21,25)(18,26,20,27)(22,28,29,24);(14,18,19,20)(15,24,21,28)(16,26,17,27)(22,23,29,25)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53576);