Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C219 |
| direct product | C22×#34411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator | C22 | C26×C4 |
| Frattini, cosocle | C23 | C27 |
| prime radical | C23×#535 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
100293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24);(9,10,12,15)(11,13,16,14)(17,19,20,24)(18,21,23,22);(3,4);(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,24);(1,2)(3,4);(5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)304 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((9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24), (9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24), (9,10,12,15)(11,13,16,14)(17,19,20,24)(18,21,23,22), (3,4), (9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,24), (1,2)(3,4), (5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23));