Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C2×D4×#442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23×D4 |
| commutator | C2×C4 | C25×C4 |
| Frattini, cosocle | C22×C4 | C26 |
| prime radical | C22×D4×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
28329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4,5,6);(4,6);(7,8,10,9)(11,12,14,18)(13,17,15,16);(12,16)(13,15)(17,18);(7,9,10,8)(11,13,14,15)(12,17,18,16)220 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (3,4,5,6), (4,6), (7,8,10,9)(11,12,14,18)(13,17,15,16), (12,16)(13,15)(17,18), (7,9,10,8)(11,13,14,15)(12,17,18,16));