Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C5 |
| prime radical | C5×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#131 ◁ C5×#504 ◁ G | |
180 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2,3,4,5)(6,7,14,21)(8,22,29,65)(9,30,34,57)(10,35,39,94)(11,20,13,18)(12,42,43,58)(15,44,49,25)(16,40,52,32)(17,53,56,23)(19,41,59,36)(24,70,74,114)(26,64,28,62)(27,80,81,115)(31,85,88,128)(33,89,90,51)(37,93,38,92)(45,99,101,123)(46,76,48,77)(47,104,105,118)(50,106,107,119)(54,68,55,69)(60,78,113,72)(61,98,103,66)(63,79,116,75)(67,83,108,109)(71,82,97,102)(73,124,125,112)(84,96,110,86)(87,91,95,127)(100,117,120,129)(111,130,131,133)(121,126,132,122);(6,8,23,66,12,27,37,76,11,26,68,104,41,79,39,44,14,29,53,98,43,81,38,77,13,28,69,118,36,75,10,25)(7,15,35,63,19,47,54,62,18,46,92,115,58,103,56,65,21,49,94,116,59,105,55,64,20,48,93,80,42,61,17,22)(9,24,67,117,33,73,95,99,40,78,106,130,96,126,88,97,34,74,108,129,90,125,87,123,32,72,119,133,86,122,31,71)(16,45,91,112,51,100,109,114,57,102,128,132,110,131,107,113,52,101,127,124,89,120,83,70,30,82,85,121,84,111,50,60);(6,9)(7,16)(8,24)(10,31)(11,40)(12,33)(13,32)(14,34)(15,45)(17,50)(18,57)(19,51)(20,30)(21,52)(22,60)(23,67)(25,71)(26,78)(27,73)(28,72)(29,74)(35,91)(36,86)(37,95)(38,87)(39,88)(41,96)(42,84)(43,90)(44,97)(46,102)(47,100)(48,82)(49,101)(53,108)(54,109)(55,83)(56,107)(58,110)(59,89)(61,111)(62,114)(63,112)(64,70)(65,113)(66,117)(68,106)(69,119)(75,122)(76,99)(77,123)(79,126)(80,121)(81,125)(85,93)(92,128)(94,127)(98,129)(103,131)(104,130)(105,120)(115,132)(116,124)(118,133)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6689);