Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13145 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,6,5)(8,13)(9,18)(10,22)(11,23)(14,24)(15,31)(16,29)(17,28)(19,21)(20,35)(25,32)(26,27)(30,34)(33,36);(3,5,6,4)(7,8,14,29,11,16,10,13,12,17,22,27,23,26,24,28)(9,15,30,38,20,31,19,25,21,33,35,37,34,36,18,32);(7,9,10,19,23,34,14,30,12,21,24,18,11,20,22,35)(8,15,13,25,26,36,29,38,17,33,28,32,16,31,27,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402091);