Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C29×C8 |
| direct product | C22×#7846 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C42 | C23 |
| commutator | C22 | C22×C43 |
| Frattini, cosocle | C2×C42 | C25 |
| socle, prime radical | C25 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
9477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(5,6,7,8)(10,13)(14,16)(18,21)(22,24);(9,10,11,14,12,13,15,16);(3,4);(17,18,19,22,20,21,23,24)448 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((1,2), (5,6,7,8)(10,13)(14,16)(18,21)(22,24), (9,10,11,14,12,13,15,16), (3,4), (17,18,19,22,20,21,23,24));