Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,11)(6,8)(7,15)(9,13)(10,14)(12,18)(16,17)(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);(4,8)(9,17)(10,15)(11,18)(19,21,23,32,25,34,24,33)(20,26,28,30,29,41,38,45)(22,31,36,46,37,47,39,48)(27,40,42,44,43,50,35,49);(3,5,6,12)(4,9,8,17)(7,16,14,13)(10,18,15,11)(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)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96482);