Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#656174 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C2×#15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
38089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(3,4,5,6)(7,8)(9,10)(12,15)(13,17)(19,20,22,24)(21,25,26,23);(8,10)(11,12,14,15)(13,16,17,18);(7,8,9,10)(12,15)(16,18)(19,21,22,26)(20,23,24,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18);(1,2);(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26)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((3,4,5,6)(7,8)(9,10)(12,15)(13,17)(19,20,22,24)(21,25,26,23), (8,10)(11,12,14,15)(13,16,17,18), (7,8,9,10)(12,15)(16,18)(19,21,22,26)(20,23,24,25), (3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18), (1,2), (11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26));