Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C218 |
| direct product | C2×#72250288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C25 | C26 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 20508 conjugacy classes of subgroups; only the 11432 normal subgroups are shown.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 26 points:
(8,10)(16,18)(19,20)(21,24)(22,23)(25,26);(15,16)(17,18)(19,21)(20,24)(22,25)(23,26);(4,6)(8,10)(12,14)(16,18)(21,26)(24,25);(4,6)(8,10)(15,16)(17,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,6)(7,8)(9,10)(11,12,13,14)(15,16)(17,18)(19,22,23,20)(21,25,26,24);(1,2)320 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,10)(16,18)(19,20)(21,24)(22,23)(25,26), (15,16)(17,18)(19,21)(20,24)(22,25)(23,26), (4,6)(8,10)(12,14)(16,18)(21,26)(24,25), (4,6)(8,10)(15,16)(17,18)(19,21)(20,25)(22,24)(23,26), (3,4)(5,6)(7,8)(9,10)(11,12,13,14)(15,16)(17,18)(19,22,23,20)(21,25,26,24), (1,2));