Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8×C11 · Schur multiplier (H2) = C29 |
| direct product | C11×#62624 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C16 | C24 |
| commutator | C2 | C24×C8×C11 |
| Frattini, cosocle | C8 | C25×C11 |
| socle | C2×C11 | C24×C8 |
| prime radical | C11×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C11×C16 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 5 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,21,17,22,31,41,18,23,32,42,33,43,53,63)(14,19,26,36,27,37,49,59,25,35,47,57,48,58,67,72)(15,20,29,39,30,40,52,62,28,38,50,60,51,61,68,73)(24,34,45,55,46,56,66,71,44,54,64,69,65,70,74,75);(12,14)(13,19)(15,24)(16,26)(17,27)(18,25)(20,34)(21,36)(22,37)(23,35)(28,44)(29,45)(30,46)(31,49)(32,47)(33,48)(38,54)(39,55)(40,56)(41,59)(42,57)(43,58)(50,64)(51,65)(52,66)(53,67)(60,69)(61,70)(62,71)(63,72)(68,74)(73,75);(12,15)(13,20)(14,24)(16,29)(17,30)(18,28)(19,34)(21,39)(22,40)(23,38)(25,44)(26,45)(27,46)(31,52)(32,50)(33,51)(35,54)(36,55)(37,56)(41,62)(42,60)(43,61)(47,64)(48,65)(49,66)(53,68)(57,69)(58,70)(59,71)(63,73)(67,74)(72,75);(15,28)(20,38)(24,44)(29,50)(30,51)(34,54)(39,60)(40,61)(45,64)(46,65)(52,68)(55,69)(56,70)(62,73)(66,74)(71,75);(14,25)(19,35)(24,44)(26,47)(27,48)(34,54)(36,57)(37,58)(45,64)(46,65)(49,67)(55,69)(56,70)(59,72)(66,74)(71,75)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 55612);