Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#3177 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,24,32)(16,26,29,41)(17,30,27,21)(20,34,35,28)(23,36,38,44)(25,40,37,33)(31,42,43,39);(6,9)(10,12)(13,15,18,24)(14,19,22,32)(16,23,29,38)(17,25,27,37)(20,31,35,43)(21,33,30,40)(26,36,41,44)(28,39,34,42);(13,16,27,34,18,29,17,28)(14,20,30,41,22,35,21,26)(15,23,37,42,24,38,25,39)(19,31,40,44,32,43,33,36);(5,7,8,11)(6,10,9,12)(15,24)(19,32)(23,38)(25,37)(31,43)(33,40)(36,44)(39,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281190);