Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C218×C4 |
| direct product | C2×#72287880 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C25×#120 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2753624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(3,4,5,6)(8,10)(12,14)(16,17)(19,20,22,24)(21,23,25,26);(19,20)(21,23)(22,24)(25,26);(7,8)(9,10)(11,12,13,14)(15,16)(17,18)(21,25)(23,26);(1,2);(12,14)(16,17)(19,21)(20,23)(22,25)(24,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,16)(19,20)(21,26)(22,24)(23,25)368 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((3,4,5,6)(8,10)(12,14)(16,17)(19,20,22,24)(21,23,25,26), (19,20)(21,23)(22,24)(25,26), (7,8)(9,10)(11,12,13,14)(15,16)(17,18)(21,25)(23,26), (1,2), (12,14)(16,17)(19,21)(20,23)(22,25)(24,26), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,16)(19,20)(21,26)(22,24)(23,25));