Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1042 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);(3,4,9,12,7,6,8,11)(5,13,15,17,16,10,14,18)(19,20,24,28,25,29,23,39)(21,30,34,41,33,45,32,26)(22,35,37,43,38,27,36,42)(31,44,48,40,47,50,46,49);(3,5,8,15,7,16,9,14)(4,10,12,17,6,13,11,18)(19,21)(20,26,29,41)(22,31)(23,32)(24,34)(25,33)(27,40,35,49)(28,30,39,45)(36,46)(37,48)(38,47)(42,50,43,44);(3,6,7,4)(5,13,16,10)(8,11,9,12)(14,17,15,18)(19,22,25,38)(20,27,29,35)(21,31,33,47)(23,37,24,36)(26,40,41,49)(28,43,39,42)(30,44,45,50)(32,48,34,46)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107362);