Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13233 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1160 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);(3,4,9,14,10,15,8,30)(5,16,20,43,21,44,31,42)(6,22,25,39,26,53,24,12)(7,27,17,40,29,13,32,34)(11,33,36,28,37,50,60,19)(18,46,48,41,49,63,57,64)(23,54,45,59,56,52,58,38)(35,47,51,61,62,55,66,65);(3,5,17,33)(4,11,34,43)(6,18,45,61)(7,28,9,31)(8,20,29,19)(10,21,32,50)(12,35,54,63)(13,16,14,36)(15,37,40,42)(22,51,38,41)(23,55,25,57)(24,48,56,47)(26,49,58,65)(27,44,30,60)(39,62,52,46)(53,66,59,64);(3,6,9,25,10,26,8,24)(4,12,30,53,15,39,14,22)(5,18,31,57,21,49,20,48)(7,23,32,58,29,56,17,45)(11,35,60,66,37,62,36,51)(13,38,34,54,27,59,40,52)(16,41,43,63,44,64,42,46)(19,47,33,61,28,55,50,65)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402396);