Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20,23,29,37,45,38,47,25,27,22,28,24,30,40,49)(21,31,33,44,41,48,36,26,35,43,32,50,34,46,39,42);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(19,21,25,35)(20,26,27,42)(22,36,23,39)(24,41,37,34)(28,46,29,48)(30,50,45,44)(31,49,43,47)(32,40,33,38);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(20,27)(26,42)(28,29)(30,45)(31,43)(44,50)(46,48)(47,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390281);