Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C222 |
| direct product | C24×#2010 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C27 |
| socle | C25 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1147544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(1,2);(3,4);(5,6)(7,11)(8,16,18,19)(9,14)(10,13)(12,17,20,15);(5,7,10,9)(6,11,13,14)(8,15,18,17)(12,16,20,19);(5,8)(6,12,13,20)(7,15)(9,17)(10,18)(11,19,14,16);(21,22);(23,24)416 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), (3,4), (5,6)(7,11)(8,16,18,19)(9,14)(10,13)(12,17,20,15), (5,7,10,9)(6,11,13,14)(8,15,18,17)(12,16,20,19), (5,8)(6,12,13,20)(7,15)(9,17)(10,18)(11,19,14,16), (21,22), (23,24));