Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C219 |
| direct product | C2×D4×C22⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C23 | C24×C42 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
970456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(4,6)(7,8,9,10)(12,14)(16,18);(4,6)(8,10)(11,12,13,14)(15,16)(17,18)(19,20,21,22);(16,18);(15,16)(17,18);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)560 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)(7,8,9,10)(12,14)(16,18), (4,6)(8,10)(11,12,13,14)(15,16)(17,18)(19,20,21,22), (16,18), (15,16)(17,18), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14));