Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217×C4 |
| direct product | C2×#72334400 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C25 | C26 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
861816 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)(25,29)(28,30);(8,10)(12,13)(17,21)(20,22)(23,24)(25,30)(26,27)(28,29);(4,6)(24,27)(28,30);(4,6)(8,10)(15,16)(17,22)(18,19)(20,21)(23,24)(25,30)(26,27)(28,29);(3,4)(5,6)(7,8)(9,10)(11,13,14,12)(15,17,18,21)(16,20,19,22)(23,25,26,29)(24,28,27,30);(1,2)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)(25,29)(28,30), (8,10)(12,13)(17,21)(20,22)(23,24)(25,30)(26,27)(28,29), (4,6)(24,27)(28,30), (4,6)(8,10)(15,16)(17,22)(18,19)(20,21)(23,24)(25,30)(26,27)(28,29), (3,4)(5,6)(7,8)(9,10)(11,13,14,12)(15,17,18,21)(16,20,19,22)(23,25,26,29)(24,28,27,30), (1,2));