Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58714 |
| 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);(4,8)(5,11)(10,15)(12,17)(35,36,41,43)(37,44,38,39)(40,48,49,42)(45,47,46,50);(4,8)(9,14)(10,15)(16,18)(20,24)(22,30)(25,27)(28,34)(35,37,41,38)(36,39,43,44)(40,45,49,46)(42,47,48,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32)(35,38,41,37)(36,39,43,44)(40,46,49,45)(42,47,48,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33)(35,39)(36,37)(38,43)(40,47)(41,44)(42,45)(46,48)(49,50);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34)(35,40)(36,42)(37,45)(38,46)(39,47)(41,49)(43,48)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012055);