Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#1581921 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
23067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(4,6)(15,16)(17,18)(19,21)(20,22)(24,27)(28,29);(4,6)(9,13)(12,14)(15,17)(16,18)(19,20)(21,22)(23,24,26,27)(25,28,30,29);(8,11)(12,14)(17,20)(18,22)(23,25,26,30)(24,28,27,29);(7,8)(9,12)(10,11)(13,14)(15,18,19,22)(16,20,21,17)(23,25)(24,28)(26,30)(27,29);(3,4)(5,6)(7,9)(8,12)(10,13)(11,14)(15,16)(17,22)(18,20)(19,21)(23,24)(25,29)(26,27)(28,30);(1,2)148 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)(15,16)(17,18)(19,21)(20,22)(24,27)(28,29), (4,6)(9,13)(12,14)(15,17)(16,18)(19,20)(21,22)(23,24,26,27)(25,28,30,29), (8,11)(12,14)(17,20)(18,22)(23,25,26,30)(24,28,27,29), (7,8)(9,12)(10,11)(13,14)(15,18,19,22)(16,20,21,17)(23,25)(24,28)(26,30)(27,29), (3,4)(5,6)(7,9)(8,12)(10,13)(11,14)(15,16)(17,22)(18,20)(19,21)(23,24)(25,29)(26,27)(28,30), (1,2));