Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#10814 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1858 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);(7,8)(9,19)(10,24,26,31)(11,17)(12,16)(13,18)(14,21)(15,25,32,27)(20,28,35,38)(22,30)(23,29)(33,34,37,36);(3,4,5,6)(7,9,12,22,13,23,11,21)(8,14,17,29,18,30,16,19)(10,20,27,36,26,35,25,34)(15,28,24,37,32,38,31,33);(7,10,11,25,13,26,12,27)(8,15,16,31,18,32,17,24)(9,20,21,34,23,35,22,36)(14,28,19,33,30,38,29,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260715);