Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4)(5,16)(6,20)(7,14)(8,13)(9,15)(10,17)(11,18)(12,23)(19,31)(21,32)(22,26)(24,30)(25,29)(27,34)(28,33);(3,5)(4,10)(6,21)(7,18)(8,25)(9,19)(11,26)(12,29)(13,27)(14,32)(15,28)(16,20)(17,23)(22,33)(24,34)(30,31);(3,5)(4,11)(6,17)(7,25)(8,18)(9,19)(10,26)(12,28)(13,31)(14,16)(15,29)(20,32)(21,23)(22,34)(24,33)(27,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26522);