Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C2×#72235099 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×#127 |
| commutator, Frattini, cosocle | C23×C4 | C26 |
| socle | C23 | C22×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
319336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(5,9)(7,10)(16,23)(19,25);(3,4)(5,7)(6,8)(9,10)(11,12,14,18,15,17,22,24)(13,19,21,23,20,25,26,16);(3,5)(4,7)(6,9)(8,10)(11,13,15,20)(12,16)(14,21,22,26)(17,23)(18,19)(24,25);(4,8)(7,10)(12,17)(16,23)(18,24)(19,25);(5,9)(7,10)(13,20)(16,23)(19,25)(21,26)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), (5,9)(7,10)(16,23)(19,25), (3,4)(5,7)(6,8)(9,10)(11,12,14,18,15,17,22,24)(13,19,21,23,20,25,26,16), (3,5)(4,7)(6,9)(8,10)(11,13,15,20)(12,16)(14,21,22,26)(17,23)(18,19)(24,25), (4,8)(7,10)(12,17)(16,23)(18,24)(19,25), (5,9)(7,10)(13,20)(16,23)(19,25)(21,26));