Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C2×#520311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator | C23 | C25×C4 |
| Frattini, cosocle | C24 | C26 |
| prime radical | C23×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
58313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(4,6)(11,12)(13,14)(15,16,20,18)(17,22,21,19);(12,14);(1,2)(7,8,10,9)(11,12)(13,14)(15,17)(16,19)(18,22)(20,21);(7,9,10,8)(11,12)(13,14)(15,16,20,18)(17,22,21,19);(3,4,5,6)(7,8,10,9)(15,18,20,16)(17,22,21,19);(15,19)(16,21)(17,18)(20,22)280 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6)(11,12)(13,14)(15,16,20,18)(17,22,21,19), (12,14), (1,2)(7,8,10,9)(11,12)(13,14)(15,17)(16,19)(18,22)(20,21), (7,9,10,8)(11,12)(13,14)(15,16,20,18)(17,22,21,19), (3,4,5,6)(7,8,10,9)(15,18,20,16)(17,22,21,19), (15,19)(16,21)(17,18)(20,22));