Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#2941965 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
33373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22);(7,9)(8,12)(10,14)(11,13)(15,17)(16,21)(18,19)(20,22);(8,10)(9,13)(16,18)(17,22)(23,24,26,28)(25,27,30,29);(4,6)(7,10,11,8)(9,14,13,12)(15,18,20,16)(17,19,22,21)(23,24)(25,29)(26,28)(27,30);(3,4)(5,6)(7,9)(8,12)(10,14)(11,13)(15,19)(16,22)(17,18)(20,21)(23,25)(24,27)(26,30)(28,29);(1,2)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((4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22), (7,9)(8,12)(10,14)(11,13)(15,17)(16,21)(18,19)(20,22), (8,10)(9,13)(16,18)(17,22)(23,24,26,28)(25,27,30,29), (4,6)(7,10,11,8)(9,14,13,12)(15,18,20,16)(17,19,22,21)(23,24)(25,29)(26,28)(27,30), (3,4)(5,6)(7,9)(8,12)(10,14)(11,13)(15,19)(16,22)(17,18)(20,21)(23,25)(24,27)(26,30)(28,29), (1,2));