Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C4×#1709 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(6,7)(8,9)(11,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,27)(26,28);(5,6,9,11,10,12,8,7)(13,15,16,22,17,23,25,28)(14,18,20,21,19,26,24,27);(5,7,8,12,10,11,9,6)(14,19)(18,26)(20,24)(21,27)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120080);