Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12712 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1781 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×2−1+4 ◁ G | |
952 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)(22,30,33,42)(23,34,36,26)(24,28)(25,29)(27,32,41,37)(31,45)(35,49,51,40)(38,43)(39,46)(44,47)(48,50);(21,26)(27,39)(28,40)(29,34)(30,44)(41,48)(42,45)(43,49);(20,22,25,33)(21,27,29,41)(23,31,36,47)(24,37,38,32)(26,39,34,48)(28,42,43,30)(35,50,51,46)(40,45,49,44)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5700);