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×#13144 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
844 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);(3,4,6,5)(8,13)(9,18)(10,14)(11,22)(15,30)(16,33)(17,27)(19,21)(20,28)(23,29)(24,31)(25,26)(32,36)(34,35);(3,5,6,4)(7,8,14,26,11,16,23,27,12,17,29,33,22,25,10,13)(9,15,28,37,20,30,18,31,21,32,35,38,34,36,19,24);(7,9,10,19,22,34,29,35,12,21,23,18,11,20,14,28)(8,15,13,24,25,36,33,38,17,32,27,31,16,30,26,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402090);