Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27225 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,28)(26,30)(27,29);(3,4,6,5)(8,11)(12,14)(15,17,18,25)(16,20,21,23)(19,24,27,26)(22,30,29,28);(3,4,6,5)(7,9,10,13)(8,12,11,14)(16,21)(17,24)(19,27)(20,28)(23,30)(25,26);(3,5,6,4)(8,11)(12,14)(16,21)(20,23)(22,29)(28,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347826);