Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33550 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C19 |
| socle | C2×C19 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C4×C19 ◁ C2×#41×C19 ◁ G | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,30)(23,27)(24,28)(25,29)(26,33)(31,42)(32,43)(34,39)(35,40)(36,41)(37,45)(38,46)(44,50)(47,49)(48,51);(20,22)(21,26)(23,32)(24,33)(25,31)(27,38)(28,30)(29,37)(34,46)(35,44)(36,45)(39,43)(40,48)(41,42)(47,51)(49,50);(22,31)(24,36)(26,37)(30,42)(32,44)(34,47)(38,48)(43,50);(20,23,25,35)(21,27,29,40)(22,32,31,44)(24,34,36,47)(26,38,37,48)(28,39,41,49)(30,43,42,50)(33,46,45,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26538);