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×#12781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1790 |
| commutator | C2×Q8 | C22×C4×C19 |
| Frattini, cosocle | C22×Q8 | C23×C19 |
| prime radical | C19×#3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C23×C19 ◁ C22×Q8×C19 ◁ G | |
758 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)(20,21,25,28,26,31,24,30)(22,32,34,50,35,46,40,49)(23,29,38,47,39,48,37,45)(27,41,43,33,44,51,42,36);(20,22)(21,27)(23,36)(24,40)(25,34)(26,35)(28,42)(29,46)(30,43)(31,44)(32,48)(33,39)(37,51)(38,41)(45,49)(47,50);(21,28)(24,25)(27,42)(29,47)(30,31)(32,49)(34,40)(37,38)(41,51)(43,44)(45,48)(46,50)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5769);