Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52826 |
| 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)(6,8)(7,14)(10,15)(11,17)(12,13)(16,18)(19,20,24,27)(21,22,29,25)(23,31,32,26)(28,33,34,30)(38,46)(41,48)(44,50)(47,49);(4,6)(9,13)(10,14)(16,17)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,17)(15,18)(19,21,24,29)(20,25,27,22)(23,28,32,34)(26,33,31,30)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,6,8,4)(5,9,12,13)(7,14,15,10)(11,16,18,17)(19,22,24,25)(20,21,27,29)(23,30,32,33)(26,28,31,34)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,7,8,15)(4,10,6,14)(5,11,12,18)(9,16,13,17)(19,23)(20,26)(21,28)(22,30)(24,32)(25,33)(27,31)(29,34)(35,38)(36,41)(37,44)(39,46)(40,47)(42,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999864);