Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#12660 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
802 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)(9,17)(10,22)(11,16)(12,14)(13,19)(15,23)(18,26)(20,27)(21,30)(24,36)(25,28)(29,32)(31,34)(33,38)(35,37);(4,6)(7,9,10,20,11,21,24,35,12,19,23,34,25,18,32,33)(8,13,15,27,16,26,29,37,14,17,22,31,28,30,36,38);(8,14)(9,18,19,21)(10,23)(13,26,17,30)(16,28)(20,35,34,33)(24,32)(27,38,31,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400304);