Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×C22⋊#401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5622 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);(4,6)(7,8,10,12)(9,11,13,14)(15,16,19,20)(17,22,21,18)(23,24)(25,28)(26,30)(27,29);(9,13)(11,14)(16,20)(17,21)(23,25,27,30)(24,28,29,26);(7,9)(8,11)(10,13)(12,14)(15,17,19,21)(16,18,20,22)(23,25)(24,26)(27,30)(28,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,18)(16,21)(17,20)(19,22)(23,26)(24,25)(27,28)(29,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514981);