Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
| direct product | C2×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | C4⋊C32 |
| prime radical | C23 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2);(3,4,7,14,8,15,26,39,9,16,27,40,30,42,53,61,10,13,28,36,31,37,54,58,22,38,48,59,49,60,64,66,11,17,29,41,32,43,55,62,19,12,33,34,46,35,56,57,6,23,24,50,25,51,52,65,5,18,20,44,21,45,47,63);(3,5,6,19,11,22,10,9)(4,12,13,18,17,16,23,38)(7,20,24,33,29,48,28,27)(8,21,25,46,32,49,31,30)(14,34,36,44,41,40,50,59)(15,35,37,45,43,42,51,60)(26,47,52,56,55,64,54,53)(39,57,58,63,62,61,65,66)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60765);