Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10313 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8,10,9)(11,12,14,13)(16,21)(18,19)(22,25)(23,24)(26,27)(29,30);(4,6)(7,9)(8,10)(11,13)(12,14)(15,16,20,24)(17,25,28,29)(18,23,19,21)(22,26,30,27);(8,12)(9,13)(11,14)(15,17,20,28)(16,22,24,30)(18,27,19,26)(21,25,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140551);