Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#7546 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 70 points:
(1,2);(3,4,5,6)(8,14)(9,20)(10,21)(11,25)(12,28)(15,24)(16,39)(17,43)(18,46)(19,34)(22,53)(23,38)(26,57)(27,52)(29,56)(30,31)(32,40)(33,44)(35,60)(36,37)(41,69)(42,64)(45,63)(47,48)(49,54)(50,51)(55,66)(58,67)(59,65)(61,70)(62,68);(7,8,15,35,10,16,36,63,25,40,66,69,51,59,54,46,12,18,38,65,26,41,22,32,29,45,48,39,52,60,9,14,13,19,20,31,27,42,47,43,56,68,53,67,57,70,23,33,28,44,49,61,50,58,55,62,11,17,37,64,21,30,24,34)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60768);