Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2223 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#466 |
| prime radical | C2×#472 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(6,12)(7,18)(8,21)(9,10)(13,22)(14,29)(15,26)(16,25)(17,27)(19,34)(20,23)(24,30)(31,36)(32,35);(5,6,13,25,10,16,8,12,11,17,21,26,9,15,22,27)(7,14,28,36,18,30,19,24,20,31,33,29,23,32,34,35);(5,7)(6,14)(8,19)(9,23)(10,18)(11,20)(12,24)(13,28)(15,32)(16,30)(17,31)(21,33)(22,34)(25,36)(26,29)(27,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403085);