Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#494889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | C24×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C22×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
5613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8)(9,19)(10,16)(11,17)(12,15)(13,18)(14,21)(20,28)(22,35)(23,32)(24,30)(25,38)(26,31)(27,34)(29,37)(33,36);(3,4,5,6)(7,9,12,21)(8,14,15,19)(10,13,24,23)(11,20,26,36)(16,32,30,18)(17,33,31,28)(22,25,37,27)(29,38,35,34);(4,6)(8,15)(9,20,21,36)(10,22,24,37)(11,25)(13,23)(14,28,19,33)(16,29,30,35)(17,34)(26,27)(31,38);(7,10,23,21,12,24,13,9)(8,16,32,14,15,30,18,19)(11,22,25,20,26,37,27,36)(17,35,38,28,31,29,34,33)160 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((1,2), (7,8)(9,19)(10,16)(11,17)(12,15)(13,18)(14,21)(20,28)(22,35)(23,32)(24,30)(25,38)(26,31)(27,34)(29,37)(33,36), (3,4,5,6)(7,9,12,21)(8,14,15,19)(10,13,24,23)(11,20,26,36)(16,32,30,18)(17,33,31,28)(22,25,37,27)(29,38,35,34), (4,6)(8,15)(9,20,21,36)(10,22,24,37)(11,25)(13,23)(14,28,19,33)(16,29,30,35)(17,34)(26,27)(31,38), (7,10,23,21,12,24,13,9)(8,16,32,14,15,30,18,19)(11,22,25,20,26,37,27,36)(17,35,38,28,31,29,34,33));