Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220×C4 |
| direct product | C23×#15674 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C25×#121 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1046280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 7 generators on 30 points:
(8,10)(12,14)(15,16)(17,21)(18,22)(19,20)(23,24,26,28)(25,27,29,30);(8,10)(12,14)(16,20)(18,21)(23,25,26,29)(24,27,28,30);(3,4);(5,6);(11,12)(13,14)(15,17,19,22)(16,18,20,21)(23,25)(24,27)(26,29)(28,30);(1,2)(3,4)(5,6);(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,21)(20,22)(23,24)(25,27)(26,28)(29,30)464 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)(17,21)(18,22)(19,20)(23,24,26,28)(25,27,29,30), (8,10)(12,14)(16,20)(18,21)(23,25,26,29)(24,27,28,30), (3,4), (5,6), (11,12)(13,14)(15,17,19,22)(16,18,20,21)(23,25)(24,27)(26,29)(28,30), (1,2)(3,4)(5,6), (7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,21)(20,22)(23,24)(25,27)(26,28)(29,30));