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×#11306 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 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);(7,8,13,16,12,17,11,18)(9,19,23,29,22,30,21,14)(10,15,26,32,27,33,25,34)(20,31,24,37,36,38,35,28);(3,4)(5,6)(7,9,13,23,12,22,11,21)(8,14,18,30,17,29,16,19)(10,24,25,20,27,35,26,36)(15,31,32,37,33,38,34,28);(4,6)(7,10,11,25,12,27,13,26)(8,15,16,32,17,33,18,34)(9,20,21,35,22,36,23,24)(14,28,19,31,29,37,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265539);