Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#33324 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C24 |
| commutator | C2 | C22×C4×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C23×C8 |
| prime radical | C24×C19 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ G | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,27,32,28,33,37,45,29,31,38,43,39,44,49,53)(26,30,35,41,36,42,48,52,34,40,46,50,47,51,54,55);(20,21,22,23)(24,26)(25,30)(27,35)(28,36)(29,34)(31,40)(32,41)(33,42)(37,48)(38,46)(39,47)(43,50)(44,51)(45,52)(49,54)(53,55);(26,34)(30,40)(35,46)(36,47)(41,50)(42,51)(48,54)(52,55);(25,31)(30,40)(32,43)(33,44)(41,50)(42,51)(45,53)(52,55)2584 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26312);