Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#3860022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C22×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
46655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(8,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,26)(22,23)(24,25);(3,4)(5,6)(7,8)(9,10)(12,16)(13,17);(3,4)(5,6)(12,16)(13,17)(19,21,24,22)(20,23,25,26);(1,2);(4,6)(11,13,15,17)(12,14,16,18)(19,22,24,21)(20,23,25,26);(3,4)(5,6)(11,14)(12,17)(13,16)(15,18)(19,23)(20,21)(22,25)(24,26)172 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((8,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,26)(22,23)(24,25), (3,4)(5,6)(7,8)(9,10)(12,16)(13,17), (3,4)(5,6)(12,16)(13,17)(19,21,24,22)(20,23,25,26), (1,2), (4,6)(11,13,15,17)(12,14,16,18)(19,22,24,21)(20,23,25,26), (3,4)(5,6)(11,14)(12,17)(13,16)(15,18)(19,23)(20,21)(22,25)(24,26));