Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7846 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)(9,14)(15,16,20,21)(17,22,24,29)(18,19,23,28)(25,30,27,26);(7,8)(9,13)(10,14)(11,12)(16,21)(17,24)(19,28)(25,27);(3,4,5,6)(7,9,11,14)(8,13,12,10)(15,17)(16,22)(18,27)(19,26)(20,24)(21,29)(23,25)(28,30);(7,10,11,13)(8,14,12,9)(15,18,20,23)(16,19,21,28)(17,25,24,27)(22,30,29,26);(15,19)(16,23)(17,26)(18,21)(20,28)(22,25)(24,30)(27,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536029);