Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20)(21,31)(22,35)(23,30)(24,29)(25,28)(26,33)(27,36)(32,40)(34,43)(37,44)(38,41)(39,42)(45,47)(46,49)(48,50);(6,15)(9,14)(11,18)(13,16)(19,21)(20,26)(22,32)(23,38)(24,33)(25,34)(27,40)(28,41)(29,43)(30,31)(35,49)(36,47)(37,48)(39,46)(42,50)(44,45);(3,5)(4,8)(6,16)(7,12)(9,18)(10,17)(11,14)(13,15)(19,22,25,37)(20,27,28,42)(21,32,34,48)(23,36,24,39)(26,40,41,50)(29,35,30,44)(31,45,43,49)(33,46,38,47);(20,28)(26,41)(27,42)(29,30)(31,43)(35,44)(40,50)(45,49)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258576);