Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×#72268042 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #1437 |
| commutator | C23×C4 | C22×C42 |
| Frattini, cosocle | C25×C4 | C24 |
| socle | C23 | #7555 |
| prime radical | C28 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C25×C4 ▷ C22 ▷ C1 | |
At least 9933 conjugacy classes of subgroups; only the 4000 normal subgroups are shown.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,14,17,15,18,23,26)(13,19,20,24,21,25,22,16);(7,10);(3,5)(4,7)(6,9)(8,10)(11,13)(12,16,18,24)(14,22)(15,21)(17,25,26,19)(20,23)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), (3,4)(5,7)(6,8)(9,10)(11,12,14,17,15,18,23,26)(13,19,20,24,21,25,22,16), (7,10), (3,5)(4,7)(6,9)(8,10)(11,13)(12,16,18,24)(14,22)(15,21)(17,25,26,19)(20,23));