Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11281 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,18)(10,22)(11,27)(12,16)(13,17)(14,20)(15,25)(19,32)(21,30)(23,37)(24,33)(26,36)(28,31)(29,38)(34,35);(7,9)(8,14)(10,23,25,38)(11,28)(12,20)(13,21)(15,32,22,34)(16,30)(17,31)(18,27)(19,26,35,24)(29,33,37,36);(7,10,11,24,13,25,12,26)(8,15,16,33,17,22,27,36)(9,19,20,23,21,35,28,38)(14,29,18,34,31,37,30,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265491);