Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C217×C4 |
| direct product | C2×#72337363 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
855192 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)(8,10)(12,14)(15,16,18,19)(17,21,22,20)(24,27)(25,29);(4,6)(7,8,9,10)(12,14)(16,19)(17,22)(23,24)(25,30)(26,27)(28,29);(11,12)(13,14)(15,16,18,19)(17,20,22,21);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(1,2);(15,17)(16,20)(18,22)(19,21)(23,25)(24,28)(26,29)(27,30)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((3,4,5,6)(8,10)(12,14)(15,16,18,19)(17,21,22,20)(24,27)(25,29), (4,6)(7,8,9,10)(12,14)(16,19)(17,22)(23,24)(25,30)(26,27)(28,29), (11,12)(13,14)(15,16,18,19)(17,20,22,21), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22), (1,2), (15,17)(16,20)(18,22)(19,21)(23,25)(24,28)(26,29)(27,30));