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×#58778 |
| 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)(5,8)(6,11)(7,12)(9,13)(10,15)(14,16)(17,18)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(4,5)(6,13)(10,12)(14,18)(22,30)(25,32)(28,34)(31,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34)(35,37,39,45)(36,40,42,43)(38,44,46,50)(41,47,48,49);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)(35,38,39,46)(36,41,42,48)(37,44,45,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012144);