Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10871 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2162 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)(9,18)(10,23)(11,27)(12,16)(13,17)(14,20)(15,24)(19,28)(21,30)(22,29)(25,31)(26,37)(32,34)(33,36)(35,38);(7,9,12,20,13,22,11,21)(8,14,16,29,17,30,27,18)(10,19,25,33,26,35,24,34)(15,28,23,36,31,38,37,32);(7,10,11,24,13,26,12,25)(8,15,16,23,17,31,27,37)(9,19,21,34,22,35,20,33)(14,28,29,36,30,38,18,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260906);