Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7472 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C23 | #366 |
| prime radical | C24 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 70 points:
(1,2);(3,4)(5,6)(7,8,11,18,15,21,14,39)(9,22,24,52,28,53,27,63)(10,19,30,47,32,49,31,64)(12,17,33,45,36,46,35,48)(13,37,34,50,38,65,29,20)(16,40,41,56,44,55,66,23)(25,51,57,68,60,69,59,43)(26,61,58,67,62,70,54,42);(4,6)(7,9,12,25,13,26,30,40,11,24,33,57,34,58,32,56,15,28,36,60,38,62,31,55,14,27,35,59,29,54,10,23)(8,16,19,42,20,43,48,63,39,66,64,70,65,69,46,53,21,44,49,67,50,68,45,52,18,41,47,61,37,51,17,22)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58963);