Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#648471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C22×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
49341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(8,9)(13,17)(16,18);(3,4,5,6)(12,15)(13,17)(19,20,24,22)(21,26,25,23);(1,2)(7,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,25)(20,23,22,26);(7,9,10,8)(13,17)(16,18)(19,20,24,22)(21,23,25,26);(3,4)(5,6)(11,13,14,17)(12,16,15,18)(19,22,24,20)(21,23,25,26);(11,12)(13,18)(14,15)(16,17)(19,23)(20,25)(21,22)(24,26)208 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((8,9)(13,17)(16,18), (3,4,5,6)(12,15)(13,17)(19,20,24,22)(21,26,25,23), (1,2)(7,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,25)(20,23,22,26), (7,9,10,8)(13,17)(16,18)(19,20,24,22)(21,23,25,26), (3,4)(5,6)(11,13,14,17)(12,16,15,18)(19,22,24,20)(21,23,25,26), (11,12)(13,18)(14,15)(16,17)(19,23)(20,25)(21,22)(24,26));