Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1028 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);(3,4,9,12,8,5,7,11)(6,10,16,18,15,13,14,17)(19,20)(21,30)(22,34,37,44)(23,38)(24,28)(25,29)(26,32)(27,36,43,35)(31,46,48,40)(33,41)(39,42)(45,47,50,49);(3,5,8,4)(6,13,15,10)(7,11,9,12)(14,17,16,18)(19,21,23,32,25,33,24,39)(20,26,28,30,29,42,38,41)(22,31,36,49,37,48,35,47)(27,40,44,45,43,46,34,50);(3,6,8,15)(4,10,5,13)(7,16,9,14)(11,18,12,17)(19,22)(20,27)(21,31)(23,36)(24,35)(25,37)(26,40)(28,44)(29,43)(30,45)(32,49)(33,48)(34,38)(39,47)(41,50)(42,46)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107967);