Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C11 |
| prime radical | C11×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 ◁ C8×C11 ◁ C11×C16 ◁ C11×C32 ◁ C11×C64 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 2 generators on 139 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(13,75)(14,138)(15,73)(16,136)(17,71)(18,134)(19,69)(20,132)(21,67)(22,130)(23,65)(24,128)(25,63)(26,126)(27,61)(28,124)(29,59)(30,122)(31,57)(32,120)(33,55)(34,118)(35,53)(36,116)(37,51)(38,114)(39,49)(40,112)(41,47)(42,110)(43,45)(44,108)(46,106)(48,104)(50,102)(52,100)(54,98)(56,96)(58,94)(60,92)(62,90)(64,88)(66,86)(68,84)(70,82)(72,80)(74,78)(77,139)(79,137)(81,135)(83,133)(85,131)(87,129)(89,127)(91,125)(93,123)(95,121)(97,119)(99,117)(101,115)(103,113)(105,111)(107,109)737 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 540);