Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11871 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,8,14,10,15,9,34)(5,16,19,42,21,47,20,55)(6,22,25,41,27,57,26,12)(7,28,31,13,33,44,32,45)(11,29,36,54,38,53,37,18)(17,48,23,58,51,65,50,46)(24,43,60,56,30,40,61,62)(35,49,63,66,64,52,39,59);(3,5,10,21)(4,11,15,38)(6,23,27,50)(7,29,33,53)(8,20,9,19)(12,39,41,63)(13,42,45,55)(14,36,34,37)(16,44,47,28)(17,26,51,25)(18,32,54,31)(22,35,57,64)(24,49,30,52)(40,46,43,58)(48,56,65,62)(59,61,66,60);(3,6,9,26,10,27,8,25)(4,12,14,22,15,41,34,57)(5,17,20,50,21,51,19,23)(7,30,31,61,33,24,32,60)(11,35,37,39,38,64,36,63)(13,43,28,62,45,40,44,56)(16,46,42,48,47,58,55,65)(18,52,29,59,54,49,53,66)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389232);