Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,13,19)(9,20,24,33)(10,14,27,30)(11,18,12,17)(15,21,34,23)(16,26,29,25)(22,32,37,31)(28,36,35,38);(8,14)(9,21)(10,25)(11,12)(15,31)(16,17)(18,29)(19,30)(20,35)(22,37)(23,24)(26,27)(28,33)(32,34);(4,6)(7,9)(8,15)(10,22)(11,21)(12,23)(13,24)(14,28)(16,32)(17,33)(18,20)(19,34)(25,36)(26,38)(27,37)(29,31)(30,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406000);