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×#57818 |
| 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 | |
4202 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);(3,4,7,10)(5,11,12,8)(6,9,14,16)(13,17,18,15)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34)(35,36)(37,39)(38,45)(40,42)(41,43)(44,47)(46,49)(48,50);(5,12)(8,11)(13,18)(15,17)(20,25)(22,26)(28,31)(30,33)(35,37,41,38)(36,39,43,45)(40,46,48,44)(42,49,50,47);(3,5)(4,8)(6,13)(7,12)(9,15)(10,11)(14,18)(16,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30)(35,38,41,37)(36,39,43,45)(40,46,48,44)(42,47,50,49);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29)(35,39)(36,38)(37,43)(40,47)(41,45)(42,46)(44,50)(48,49);(3,6,7,14)(4,9,10,16)(5,13,12,18)(8,15,11,17)(19,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,33)(27,34)(35,40,41,48)(36,42,43,50)(37,44,38,46)(39,47,45,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008947);