Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8633 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
814 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);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16)(19,20)(21,30)(22,35,37,42)(23,38)(24,28)(25,29)(26,32)(27,36,43,39)(31,40,47,50)(33,41)(34,45)(44,46,49,48);(19,21,24,33,25,34,23,32)(20,26,38,45,29,41,28,30)(22,31,39,48,37,47,36,46)(27,40,35,49,43,50,42,44);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)(19,22)(20,27)(21,31)(23,36)(24,39)(25,37)(26,40)(28,42)(29,43)(30,44)(32,46)(33,48)(34,47)(35,38)(41,50)(45,49)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96180);