Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#17076 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8)(9,13)(10,11)(12,14)(16,18)(17,21)(23,24,28,30)(25,29,27,26);(7,9)(8,12)(10,13)(11,14)(15,16,19,18)(17,22,21,20)(23,25,28,27)(24,26,30,29);(7,9)(8,13)(10,12)(11,14)(15,17,19,21)(16,20,18,22)(23,26)(24,25)(27,30)(28,29);(3,4,5,6)(7,10,11,8)(9,13,14,12)(15,18,19,16)(17,20,21,22)(23,27,28,25)(24,29,30,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519487);