Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C42 |
| direct product | C2×#918022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
39673 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)(11,12,16,18)(13,15,14,17)(20,24)(22,25);(7,8)(9,10)(11,13,16,14)(12,17,18,15)(19,20)(21,22)(23,24)(25,26);(19,21)(20,25)(22,24)(23,26);(1,2);(3,4,5,6)(11,14,16,13)(12,15,18,17)(19,21)(20,22)(23,26)(24,25);(3,4)(5,6)(11,15)(12,14)(13,18)(16,17)(19,22)(20,26)(21,24)(23,25)160 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)(11,12,16,18)(13,15,14,17)(20,24)(22,25), (7,8)(9,10)(11,13,16,14)(12,17,18,15)(19,20)(21,22)(23,24)(25,26), (19,21)(20,25)(22,24)(23,26), (1,2), (3,4,5,6)(11,14,16,13)(12,15,18,17)(19,21)(20,22)(23,26)(24,25), (3,4)(5,6)(11,15)(12,14)(13,18)(16,17)(19,22)(20,26)(21,24)(23,25));