Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#14994 |
| 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 | |
3294 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);(7,8,10,12)(9,13,14,11)(17,21)(19,22)(23,24,27,30)(25,29,26,28);(4,6)(15,16,18,20)(17,19,21,22)(23,25,27,26)(24,28,30,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26,27,25)(24,29,30,28);(7,9,10,14)(8,11,12,13)(15,17)(16,19)(18,21)(20,22)(23,24)(25,28)(26,29)(27,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515239);