Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216×C4 |
| direct product | C2×#72334398 |
| 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 | |
At least 6676 conjugacy classes of subgroups; only the 10744 normal subgroups are shown.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(8,10)(12,14)(15,16)(17,18)(19,20)(21,22)(24,27)(28,29);(8,10)(16,20)(18,22)(23,24)(25,29)(26,27)(28,30);(4,6)(8,10)(12,14)(24,27)(28,29);(4,6)(8,10)(15,17)(16,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,30);(3,4)(5,6)(7,8)(9,10)(11,12,13,14)(15,18)(16,21)(17,20)(19,22)(23,25,26,30)(24,28,27,29);(1,2)296 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((8,10)(12,14)(15,16)(17,18)(19,20)(21,22)(24,27)(28,29), (8,10)(16,20)(18,22)(23,24)(25,29)(26,27)(28,30), (4,6)(8,10)(12,14)(24,27)(28,29), (4,6)(8,10)(15,17)(16,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,30), (3,4)(5,6)(7,8)(9,10)(11,12,13,14)(15,18)(16,21)(17,20)(19,22)(23,25,26,30)(24,28,27,29), (1,2));