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×#11350 |
| 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)(9,17)(10,14)(11,24)(12,16)(13,20)(15,18)(19,32)(21,33)(22,31)(23,30)(25,29)(26,37)(27,34)(28,35)(36,38);(7,9,18,34,12,21,11,20)(8,13,24,33,16,27,15,17)(10,19,22,35,23,36,25,37)(14,28,29,32,30,26,31,38);(7,10,18,22,12,23,11,25)(8,14,24,29,16,30,15,31)(9,19,34,35,21,36,20,37)(13,26,33,38,27,28,17,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265803);