Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C19 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×#41×C19 ◁ G | |
1108 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)(22,33)(23,31)(24,30)(25,40)(26,42)(27,32)(28,34)(29,38)(35,43)(36,50)(37,48)(39,47)(41,46)(44,49)(45,51);(20,22)(21,28)(23,34)(24,36)(25,41)(26,35)(27,37)(29,43)(30,45)(31,48)(32,46)(33,40)(38,51)(39,49)(42,44)(47,50);(20,23,27,25)(21,29,32,47)(22,34,37,41)(24,38,26,39)(28,43,46,50)(30,40,42,31)(33,44,48,45)(35,49,36,51);(20,24,27,26)(21,30,32,42)(22,35,37,36)(23,38,25,39)(28,44,46,45)(29,40,47,31)(33,43,48,50)(34,49,41,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26560);