Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8454 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)(8,12)(13,14)(15,16)(17,24)(18,21)(19,22)(20,23)(25,30)(26,28)(27,29);(7,8)(9,13)(10,14)(11,12)(15,17,21,19)(16,22,18,24)(20,25,28,27)(23,29,26,30);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,18,21,16)(17,22,19,24)(20,26,28,23)(25,29,27,30);(7,9,11,10)(8,13,12,14)(15,19,21,17)(16,22,18,24)(20,25,28,27)(23,30,26,29);(7,10,11,9)(8,14,12,13)(15,20,21,28)(16,23,18,26)(17,25,19,27)(22,29,24,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536780);