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×#54576 |
| 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 | |
4132 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);(3,4)(5,9)(6,14)(7,8)(10,15)(11,12)(13,18)(16,17)(19,20,24,25)(21,22,26,29)(23,31,32,27)(28,34,33,30)(35,36,40,43)(37,41,38,45)(39,42,47,50)(44,48,46,49);(4,8)(5,11)(6,15)(16,18)(20,25)(22,29)(27,31)(30,34)(35,37,40,38)(36,41,43,45)(39,46,47,44)(42,49,50,48);(3,5,7,11)(4,9,8,12)(6,13,15,17)(10,16,14,18)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34)(35,38,40,37)(36,41,43,45)(39,46,47,44)(42,48,50,49);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,49)(47,50);(3,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)(35,39,40,47)(36,42,43,50)(37,44,38,46)(41,48,45,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011060);