Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37281 |
| 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 | |
7264 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);(4,6)(9,13)(12,14)(15,16,21,24)(17,19,18,22)(20,28,29,23)(25,30,26,27);(8,11)(12,14)(15,17,21,18)(16,22,24,19)(20,25,29,26)(23,30,28,27);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18,21,17)(16,22,24,19)(20,26,29,25)(23,30,28,27);(7,8)(9,12)(10,11)(13,14)(15,19,21,22)(16,17,24,18)(20,27,29,30)(23,25,28,26);(7,9,10,13)(8,12,11,14)(15,20,21,29)(16,23,24,28)(17,25,18,26)(19,27,22,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541924);