Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2176 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)(5,12,14,8)(6,11)(7,10)(9,13,17,15)(16,18)(19,20,24,29,25,30,23,28)(21,31,35,26,34,41,33,40)(22,27,36,42,37,43,38,44)(32,46,47,50,48,45,49,39);(3,5)(4,8)(6,13)(7,14)(9,11)(10,12)(15,16)(17,18)(19,21)(20,26)(22,32)(23,35)(24,33)(25,34)(27,39)(28,41)(29,31)(30,40)(36,49)(37,48)(38,47)(42,45)(43,50)(44,46);(3,5)(4,9)(6,15)(7,14)(8,11)(10,17)(12,18)(13,16)(19,22,25,37)(20,27,30,43)(21,32,34,48)(23,38,24,36)(26,39,40,50)(28,44,29,42)(31,45,41,46)(33,49,35,47)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94703);