Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(5,11)(9,14)(12,17)(16,18)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34)(35,36,40,38)(37,43,45,41)(39,42,47,46)(44,49,50,48);(4,8)(6,13)(9,14)(12,17)(20,21)(22,29)(26,28)(30,34)(39,47)(42,46)(44,50)(48,49);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32)(35,37,40,45)(36,41,38,43)(39,44,47,50)(42,48,46,49);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,38,40,36)(37,43,45,41)(39,46,47,42)(44,49,50,48);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)(35,39)(36,42)(37,44)(38,46)(40,47)(41,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013104);