Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1790 |
| commutator | C2×Q8 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×#41×C19 ◁ G | |
756 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)(22,32)(23,36,24,29)(25,30)(26,31)(27,33,44,38)(28,37,45,39)(34,48)(35,46)(40,42)(41,47)(43,49,51,50);(20,22)(21,27)(23,33)(24,38)(25,40)(26,35)(28,42)(29,46)(30,43)(31,44)(32,36)(34,41)(37,50)(39,49)(45,48)(47,51);(20,23,26,24)(21,28,31,45)(22,33,35,38)(25,39,41,37)(27,42,44,48)(29,47,36,30)(32,43,46,51)(34,50,40,49)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5768);