Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215×C43 |
| direct product | C2×#72339321 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C25×#127 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
1024776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(3,4)(5,6)(7,8)(9,10)(11,12,13,14)(16,19)(17,21)(23,24,26,27)(25,29,30,28);(8,10)(16,19)(20,22)(24,27)(28,29);(4,5)(8,10)(12,14)(23,24,26,27)(25,28,30,29);(3,5,6,4)(7,8)(9,10)(11,12)(13,14)(15,16,18,19)(17,22,21,20)(23,24)(25,29)(26,27)(28,30);(1,2);(15,17)(16,20)(18,21)(19,22)(23,25)(24,28)(26,30)(27,29)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((3,4)(5,6)(7,8)(9,10)(11,12,13,14)(16,19)(17,21)(23,24,26,27)(25,29,30,28), (8,10)(16,19)(20,22)(24,27)(28,29), (4,5)(8,10)(12,14)(23,24,26,27)(25,28,30,29), (3,5,6,4)(7,8)(9,10)(11,12)(13,14)(15,16,18,19)(17,22,21,20)(23,24)(25,29)(26,27)(28,30), (1,2), (15,17)(16,20)(18,21)(19,22)(23,25)(24,28)(26,30)(27,29));