Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C2×#763960 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C23×#534 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
57141 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)(8,10)(11,12,14,16)(13,17,18,15)(20,22)(24,25);(7,8)(9,10)(13,18)(15,17)(19,20)(21,25)(22,23)(24,26);(19,21)(20,24)(22,25)(23,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,24)(23,26);(11,13,14,18)(12,15,16,17)(19,22,23,20)(21,25,26,24)184 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)(8,10)(11,12,14,16)(13,17,18,15)(20,22)(24,25), (7,8)(9,10)(13,18)(15,17)(19,20)(21,25)(22,23)(24,26), (19,21)(20,24)(22,25)(23,26), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,24)(23,26), (11,13,14,18)(12,15,16,17)(19,22,23,20)(21,25,26,24));