Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9765 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| 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 | |
956 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);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9)(19,20)(21,30,33,41)(22,34)(23,28)(24,39)(25,29)(26,32,42,38)(27,36)(31,40,46,50)(35,43)(37,48)(44,49,47,45);(3,5,8,13)(4,9)(6,14,7,15)(10,12)(11,17)(16,18)(19,21,23,38,25,33,24,32)(20,26,28,41,29,42,39,30)(22,31,35,49,37,46,36,45)(27,40,34,47,43,50,48,44);(5,13)(9,17)(12,18)(14,15)(19,22,24,36,25,37,23,35)(20,27,28,34,29,43,39,48)(21,31,32,45,33,46,38,49)(26,40,41,47,42,50,30,44)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112780);