Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1786 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
928 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)(22,32,35,47)(23,37)(24,29)(25,31)(26,33,43,50)(27,34,44,36)(28,39)(30,41)(38,48,51,42)(40,45)(46,49);(20,22)(21,26)(23,38)(24,34)(25,35)(27,28)(29,42)(30,48)(31,43)(32,45)(33,49)(36,41)(37,47)(39,50)(40,51)(44,46);(20,22)(21,27)(23,33)(24,36)(25,35)(26,28)(29,47)(30,32)(31,44)(34,41)(37,42)(38,49)(39,51)(40,50)(43,46)(45,48)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5738);