Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C218×C4 |
| direct product | C2×#72287890 |
| 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 | |
2755288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(4,6)(8,9)(12,13)(15,16,17,18)(20,23)(24,26);(3,4)(5,6)(7,8,10,9)(11,12)(13,14)(15,16)(17,18)(21,25)(24,26);(12,13)(15,16)(17,18);(1,2);(3,4)(5,6)(7,9,10,8)(11,13,14,12)(15,16)(17,18)(19,20)(21,26)(22,23)(24,25);(19,21)(20,24)(22,25)(23,26)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((4,6)(8,9)(12,13)(15,16,17,18)(20,23)(24,26), (3,4)(5,6)(7,8,10,9)(11,12)(13,14)(15,16)(17,18)(21,25)(24,26), (12,13)(15,16)(17,18), (1,2), (3,4)(5,6)(7,9,10,8)(11,13,14,12)(15,16)(17,18)(19,20)(21,26)(22,23)(24,25), (19,21)(20,24)(22,25)(23,26));