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×#36031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8470 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)(7,8,11,10)(9,13,14,12)(15,16,20,23)(17,21,24,28)(18,19,22,27)(25,26,29,30);(7,9,11,14)(8,12,10,13)(17,24)(21,28)(25,29)(26,30);(3,4)(5,6)(7,10,11,8)(9,13,14,12)(15,17)(16,21)(18,25)(19,26)(20,24)(22,29)(23,28)(27,30);(15,18)(16,19)(17,25)(20,22)(21,26)(23,27)(24,29)(28,30);(15,19)(16,22)(17,26)(18,23)(20,27)(21,29)(24,30)(25,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536650);