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 | C2×#17972 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4,8,11,7,10,16,18)(5,12,14,15,13,9,17,6)(19,20)(21,29)(22,26)(23,35)(24,27)(25,31)(28,37)(30,43)(32,42)(33,41)(34,47)(36,39)(38,45)(40,46)(44,50)(48,49);(3,5)(4,9)(6,11)(7,13)(8,14)(10,12)(15,18)(16,17)(19,21,24,32)(20,25,27,39)(22,30,33,44)(23,36,37,31)(26,38,41,49)(28,42,35,29)(34,48,46,45)(40,50,47,43);(3,6,8,12,7,15,16,9)(4,5,11,14,10,13,18,17)(19,22,24,33)(20,26,27,41)(21,30,32,44)(23,34,37,46)(25,38,39,49)(28,40,35,47)(29,43,42,50)(31,45,36,48);(4,10)(5,13)(11,18)(14,17)(20,27)(22,33)(25,39)(28,35)(29,42)(30,44)(34,46)(45,48)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251997);