Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217 |
| direct product | C2×#72334407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C25 | C26 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
837792 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)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,27,30,29);(7,8)(9,10)(16,17)(20,21);(11,12)(13,14)(23,24,26,28)(25,27,30,29);(4,6)(8,10)(11,12,13,14)(15,17,19,16)(18,20,22,21)(23,24)(25,29)(26,28)(27,30);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,21)(19,22)(23,25)(24,27)(26,30)(28,29)416 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)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,27,30,29), (7,8)(9,10)(16,17)(20,21), (11,12)(13,14)(23,24,26,28)(25,27,30,29), (4,6)(8,10)(11,12,13,14)(15,17,19,16)(18,20,22,21)(23,24)(25,29)(26,28)(27,30), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,21)(19,22)(23,25)(24,27)(26,30)(28,29));