Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#1192577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
860336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(13,14,16,15)(17,18,20,22)(19,23,24,21);(3,4);(5,6)(7,8)(10,12)(14,15);(1,2)(3,4);(6,7)(19,24)(21,23);(5,6,8,7)(9,10,11,12)(13,15,16,14)(17,18)(19,21)(20,22)(23,24);(5,7,8,6)(17,19,20,24)(18,21,22,23)344 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((13,14,16,15)(17,18,20,22)(19,23,24,21), (3,4), (5,6)(7,8)(10,12)(14,15), (1,2)(3,4), (6,7)(19,24)(21,23), (5,6,8,7)(9,10,11,12)(13,15,16,14)(17,18)(19,21)(20,22)(23,24), (5,7,8,6)(17,19,20,24)(18,21,22,23));