Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#12577 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#99 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | C2×#342 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×#41×C19 ◁ G | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 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)(23,34,35,27)(24,28)(25,29)(26,32,38,44)(31,42)(33,43)(36,47,48,40)(37,41)(39,46,49,51)(45,50);(20,22,24,33,25,31,37,45)(21,26,28,39,29,38,41,49)(23,32,36,46,35,44,48,51)(27,30,40,43,34,42,47,50);(22,31)(26,38)(30,42)(32,44)(33,45)(39,49)(43,50)(46,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5565);