Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C215 |
| direct product | C2×#72237261 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×#127 |
| commutator, Frattini, cosocle | C24×C4 | C25 |
| socle | C24 | C2×#392 |
| LCS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
314840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(7,8)(9,10)(17,19)(21,26);(3,4)(5,6)(7,9,10,8)(11,12,16,13)(14,21)(15,18,24,20)(17,25)(19,22)(23,26);(8,9)(11,13,16,12)(14,21)(15,20,24,18)(17,25)(19,22)(23,26);(4,6)(11,14,16,23)(12,17)(13,19)(15,22,24,25)(18,21)(20,26)272 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((1,2), (7,8)(9,10)(17,19)(21,26), (3,4)(5,6)(7,9,10,8)(11,12,16,13)(14,21)(15,18,24,20)(17,25)(19,22)(23,26), (8,9)(11,13,16,12)(14,21)(15,20,24,18)(17,25)(19,22)(23,26), (4,6)(11,14,16,23)(12,17)(13,19)(15,22,24,25)(18,21)(20,26));