Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#2253 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4);(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,23,25,37)(17,30,28,43)(18,31,29,22)(20,34,32,26)(21,36,33,24)(27,42,41,39)(35,40,38,44);(5,7,8,12)(6,9,10,11)(14,19)(20,32)(21,33)(22,31)(23,37)(30,43)(35,38)(39,42);(13,15,24,40,18,27,28,34,16,25,36,44,29,41,17,26)(14,20,30,39,22,35,33,37,19,32,43,42,31,38,21,23)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403345);