Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12604 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,35)(27,40)(28,42)(29,32)(30,34)(31,37)(33,44)(36,48)(38,52)(39,45)(41,47)(43,54)(46,53)(49,50)(51,55);(24,26,28,38,29,39,43,53,30,36,44,50,27,37,41,51)(25,31,33,46,32,35,47,49,34,45,42,55,40,48,54,52);(20,22,23,21)(26,36)(31,45)(35,48)(37,39)(38,50)(46,55)(49,52)(51,53)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5592);