Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46227 |
| 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 | |
7210 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)(8,9)(12,13)(15,16,20,24)(17,23,19,21)(18,22,27,30)(25,29,26,28);(8,9)(10,14)(15,17,20,19)(16,21,24,23)(18,26,27,25)(22,29,30,28);(3,4)(5,6)(7,8,11,9)(10,12,14,13)(15,16)(17,21)(18,22)(19,23)(20,24)(25,28)(26,29)(27,30);(7,9,11,8)(10,13,14,12)(15,18,20,27)(16,22,24,30)(17,25,19,26)(21,28,23,29);(7,10)(8,12)(9,13)(11,14)(15,19,20,17)(16,23,24,21)(18,26,27,25)(22,29,30,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610111);