Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C227 |
| direct product | C2×#72334969 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator | C24 | C26×C4 |
| Frattini, cosocle | C25 | C27 |
| prime radical | C28×D4 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
226594363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 26 points:
(12,14)(16,18);(1,2);(4,6)(8,10)(12,14)(16,18)(19,20,23,22)(21,24,25,26);(4,6)(8,10)(12,14)(20,22)(24,26);(12,14);(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,16)(17,18)(19,22,23,20)(21,24,25,26)832 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((12,14)(16,18), (1,2), (4,6)(8,10)(12,14)(16,18)(19,20,23,22)(21,24,25,26), (4,6)(8,10)(12,14)(20,22)(24,26), (12,14), (19,21)(20,24)(22,26)(23,25), (3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,16)(17,18)(19,22,23,20)(21,24,25,26));