Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C24×C5 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 ◁ C5×#131 ◁ C5×#504 ◁ G | |
452 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 133 points:
(1,2,3,4,5)(6,7,13,19)(8,20,25,62)(9,15,26,47)(10,30,35,59)(11,36,39,92)(12,40,38,18)(14,42,46,22)(16,51,54,33)(17,55,57,32)(21,60,63,112)(23,68,69,109)(24,70,41,61)(27,74,71,111)(28,79,72,120)(29,81,73,50)(31,56,87,110)(34,90,89,84)(37,52,93,108)(43,94,95,64)(44,91,98,86)(45,67,58,85)(48,101,99,77)(49,104,100,76)(53,88,83,107)(65,116,114,133)(66,117,82,113)(75,105,118,131)(78,127,119,123)(80,102,121,130)(96,128,129,125)(97,115,106,124)(103,126,122,132);(6,8,13,25)(7,14,19,46)(9,21,26,63)(10,31,35,87)(11,37,39,93)(12,41,38,24)(15,43,47,95)(16,52,54,108)(17,56,57,110)(18,58,40,45)(20,59,62,30)(22,51,42,33)(23,55,69,32)(27,75,71,118)(28,80,72,121)(29,82,73,66)(34,91,89,86)(36,44,92,98)(48,102,99,130)(49,105,100,131)(50,106,81,97)(53,109,83,68)(60,111,112,74)(61,84,70,90)(64,101,94,77)(65,104,114,76)(67,88,85,107)(78,128,119,125)(79,96,120,129)(103,133,122,116)(113,123,117,127)(115,126,124,132);(6,9)(7,15)(8,21)(10,27)(11,28)(12,29)(13,26)(14,43)(16,48)(17,49)(18,50)(19,47)(20,60)(22,64)(23,65)(24,66)(25,63)(30,74)(31,75)(32,76)(33,77)(34,78)(35,71)(36,79)(37,80)(38,73)(39,72)(40,81)(41,82)(42,94)(44,96)(45,97)(46,95)(51,101)(52,102)(53,103)(54,99)(55,104)(56,105)(57,100)(58,106)(59,111)(61,113)(62,112)(67,115)(68,116)(69,114)(70,117)(83,122)(84,123)(85,124)(86,125)(87,118)(88,126)(89,119)(90,127)(91,128)(92,120)(93,121)(98,129)(107,132)(108,130)(109,133)(110,131);(9,26)(15,47)(21,63)(27,71)(28,72)(29,73)(43,95)(48,99)(49,100)(50,81)(60,112)(64,94)(65,114)(66,82)(74,111)(75,118)(76,104)(77,101)(78,119)(79,120)(80,121)(96,129)(97,106)(102,130)(103,122)(105,131)(113,117)(115,124)(116,133)(123,127)(125,128)(126,132)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26972);