Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C29 |
| direct product | C11×#62637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C11×#41 ◁ C2×C11×#131 ◁ G | |
2026 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)(14,24)(15,20)(16,32)(17,37)(18,21)(19,26)(22,35)(23,34)(25,49)(27,56)(28,50)(29,42)(30,59)(31,62)(33,47)(36,48)(38,46)(39,52)(40,55)(41,64)(43,61)(44,58)(45,54)(51,71)(53,74)(57,70)(60,69)(63,68)(65,72)(66,75)(67,73);(12,14,18,28)(13,19,21,40)(15,25,29,51)(16,33,36,64)(17,38,34,27)(20,39,42,65)(22,46,45,56)(23,47,37,41)(24,48,50,32)(26,54,55,35)(30,60,57,75)(31,63,58,53)(43,68,67,74)(44,69,62,66)(49,70,71,59)(52,73,72,61);(12,15)(13,20)(14,25)(16,30)(17,31)(18,29)(19,39)(21,42)(22,43)(23,44)(24,49)(26,52)(27,53)(28,51)(32,59)(33,60)(34,58)(35,61)(36,57)(37,62)(38,63)(40,65)(41,66)(45,67)(46,68)(47,69)(48,70)(50,71)(54,73)(55,72)(56,74)(64,75);(15,29)(20,42)(25,51)(30,57)(31,58)(39,65)(43,67)(44,62)(49,71)(52,72)(53,63)(59,70)(60,75)(61,73)(66,69)(68,74);(13,21)(19,40)(20,42)(22,45)(23,37)(24,50)(32,48)(39,65)(41,47)(43,67)(44,62)(46,56)(49,71)(59,70)(66,69)(68,74)638 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 55625);