Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12713 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1781 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×H(5,𝔽2) ◁ G | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,21,23,14)(10,15)(11,16)(12,18,28,33)(13,22,29,24)(19,26)(20,31)(25,30)(27,32,37,34)(35,36);(6,8,11,20)(7,12,16,28)(9,18,23,33)(10,19,25,35)(13,26,29,36)(14,17,21,31)(15,27,30,37)(22,32,24,34);(6,9)(7,13)(8,18)(10,24)(11,23)(12,26)(14,15)(16,29)(17,27)(19,34)(20,33)(21,30)(22,25)(28,36)(31,37)(32,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5701);