Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C43 |
| direct product | C2×#4452721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
35529 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)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(21,25)(23,26);(4,6)(8,9)(13,18)(15,17)(19,20,22,24)(21,26,25,23);(19,20)(21,23)(22,24)(25,26);(1,2);(3,4,5,6)(7,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,20)(21,26)(22,24)(23,25);(11,13,14,18)(12,15,16,17)(19,21,22,25)(20,23,24,26)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((4,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(21,25)(23,26), (4,6)(8,9)(13,18)(15,17)(19,20,22,24)(21,26,25,23), (19,20)(21,23)(22,24)(25,26), (1,2), (3,4,5,6)(7,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,20)(21,26)(22,24)(23,25), (11,13,14,18)(12,15,16,17)(19,21,22,25)(20,23,24,26));