Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×D128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C11 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 ◁ C8×C11 ◁ C11×C16 ◁ C11×C32 ◁ C11×C64 ◁ G | |
46 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);(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)737 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 539);