Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#20422 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,16,13,17,11,26)(9,18,22,28,20,32,21,14)(10,15,25,30,23,29,24,36)(19,31,34,37,35,38,33,27);(4,6)(7,9)(8,14,17,28)(10,19)(11,21)(12,22)(13,20)(15,27,29,37)(16,18,26,32)(23,35)(24,33)(25,34)(30,31,36,38);(7,10,12,25,13,23,11,24)(8,15,16,30,17,29,26,36)(9,19,22,34,20,35,21,33)(14,27,18,31,28,37,32,38);(9,20)(10,23)(14,28)(15,29)(18,32)(21,22)(24,25)(30,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277061);