Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#612519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C2×#14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
41579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(4,5)(7,8)(9,10)(11,12,15,13)(14,16,18,17)(19,20,23,24)(21,25,26,22);(3,4,6,5)(7,8,9,10)(12,13)(14,18)(20,24)(22,25);(1,2)(3,5,6,4)(7,8)(9,10)(11,13,15,12)(14,16,18,17)(19,21,23,26)(20,22,24,25);(3,4,6,5)(7,8)(9,10)(12,13)(14,18)(19,22)(20,21)(23,25)(24,26);(3,5,6,4)(11,14)(12,16)(13,17)(15,18);(11,12)(13,15)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24)232 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((4,5)(7,8)(9,10)(11,12,15,13)(14,16,18,17)(19,20,23,24)(21,25,26,22), (3,4,6,5)(7,8,9,10)(12,13)(14,18)(20,24)(22,25), (1,2)(3,5,6,4)(7,8)(9,10)(11,13,15,12)(14,16,18,17)(19,21,23,26)(20,22,24,25), (3,4,6,5)(7,8)(9,10)(12,13)(14,18)(19,22)(20,21)(23,25)(24,26), (3,5,6,4)(11,14)(12,16)(13,17)(15,18), (11,12)(13,15)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24));