Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7229 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C22 | #1289 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
600 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);(3,4,7,12,9,14,21,37)(5,15,17,39,19,41,42,58)(6,20,22,11,25,34,24,36)(8,27,26,38,28,56,50,13)(10,29,30,16,33,43,32,45)(18,47,46,62,48,66,63,40)(23,52,51,49,54,35,53,61)(31,44,59,64,60,65,55,57);(3,5,8,18,21,42,50,63,9,19,28,48,7,17,26,46)(4,10,13,31,12,32,27,55,14,33,38,60,37,30,56,59)(6,16,23,44,22,43,51,64,25,45,54,65,24,29,53,57)(11,15,35,40,20,39,49,47,36,41,52,62,34,58,61,66)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34515);