Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1490 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);(5,6,10,14,11,15,9,22)(7,16,18,12,19,27,23,26)(8,13,20,29,21,30,24,28)(17,32,35,36,34,31,33,25);(5,7,10,18,11,19,9,23)(6,12,14,27,15,26,22,16)(8,17,20,35,21,34,24,33)(13,25,29,32,30,36,28,31);(5,8)(6,13)(7,17)(9,24)(10,20)(11,21)(12,25)(14,29)(15,30)(16,31)(18,35)(19,34)(22,28)(23,33)(26,36)(27,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261650);