Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
| direct product | C16×Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle, prime radical | C22 | C8×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34);(17,33,25,41)(18,34,26,42)(19,35,27,43)(20,36,28,44)(21,37,29,45)(22,38,30,46)(23,39,31,47)(24,40,32,48)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30139);