Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C220 |
| direct product | C2×#72287883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C23 | C24×C42 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C29 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2852552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(7,8)(9,10)(15,16,18,17)(20,23)(24,25);(19,20)(21,24)(22,23)(25,26);(4,6)(8,10)(11,12,13,14)(15,17,18,16)(19,20)(21,25)(22,23)(24,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14);(19,21)(20,24)(22,26)(23,25)512 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((7,8)(9,10)(15,16,18,17)(20,23)(24,25), (19,20)(21,24)(22,23)(25,26), (4,6)(8,10)(11,12,13,14)(15,17,18,16)(19,20)(21,25)(22,23)(24,26), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14), (19,21)(20,24)(22,26)(23,25));