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×#54220 |
| 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 | |
4170 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)(19,20,24,28)(21,25,29,34)(22,26,32,31)(23,27,33,30)(35,36,40,43)(37,38,45,41)(39,47,48,42)(44,49,50,46);(4,8)(5,11)(6,15)(16,18)(21,29)(23,33)(25,34)(27,30)(35,37,40,45)(36,41,43,38)(39,44,48,50)(42,49,47,46);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21)(20,25)(22,27)(23,31)(24,29)(26,33)(28,34)(30,32)(35,37)(36,41)(38,43)(39,44)(40,45)(42,49)(46,47)(48,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,18)(15,17)(19,22)(20,26)(21,30)(23,25)(24,32)(27,29)(28,31)(33,34)(35,38,40,41)(36,37,43,45)(39,46,48,49)(42,44,47,50);(3,6,7,15)(4,10,8,14)(5,13,11,17)(9,16,12,18)(19,23)(20,27)(21,31)(22,25)(24,33)(26,29)(28,30)(32,34)(35,39,40,48)(36,42,43,47)(37,44,45,50)(38,46,41,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010115);