Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7208 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C22 | #1282 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
608 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);(3,4,7,13,9,15,28,38,10,14,8,29,27,39,30,55)(5,16,18,40,19,43,47,63,20,42,25,41,48,62,52,64)(6,21,22,36,23,49,50,61,24,12,26,35,51,37,53,60)(11,31,32,45,34,57,58,46,33,44,54,56,59,65,66,17);(3,5,8,18,9,19,30,47,10,20,7,25,27,48,28,52)(4,11,13,32,15,34,38,58,14,33,29,54,39,59,55,66)(6,17,22,44,23,45,50,65,24,46,26,31,51,56,53,57)(12,16,36,41,37,43,61,64,21,42,35,40,49,62,60,63)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34279);