Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,11)(10,12)(13,14)(16,21)(17,24)(19,27)(26,29);(7,8)(9,13)(10,12)(11,14)(15,16,20,21)(17,18,24,22)(19,28,27,23)(25,29,30,26);(3,4)(5,6)(7,8,10,12)(9,11,14,13)(15,17,20,24)(16,22,21,18)(19,25,27,30)(23,29,28,26);(7,8)(9,11)(10,12)(13,14)(15,18,20,22)(16,17,21,24)(19,26,27,29)(23,25,28,30);(7,9)(8,11)(10,14)(12,13)(15,19)(16,23)(17,25)(18,26)(20,27)(21,28)(22,29)(24,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608097);