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×#7473 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C23 | #357 |
| prime radical | C24 | SD32 |
| 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 | |
386 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)(9,21)(10,27)(11,33)(12,36)(13,37)(14,20)(15,22)(16,30)(17,43)(18,28)(19,29)(23,56)(24,38)(25,59)(26,50)(31,46)(32,45)(34,63)(35,60)(39,58)(40,67)(41,55)(42,62)(44,61)(47,53)(48,51)(49,52)(54,64)(57,66)(65,69)(68,70);(7,9,11,23,34,57,10,22,29,52,62,70,28,51,43,67,14,26,35,58,12,24,32,55,13,25,30,53,61,69,31,54)(8,15,17,38,44,56,42,50,37,64,27,48,36,47,33,49,20,41,46,66,18,39,16,21,19,40,45,65,63,68,60,59)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58967);