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×#56181 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20)(21,27)(22,24)(23,30)(25,29)(26,31)(28,32)(33,34)(35,36)(37,44)(38,41)(39,47)(40,43)(42,48)(45,49)(46,50);(4,5)(6,13)(7,15)(16,17)(20,22)(23,31)(25,27)(28,34)(35,37,40,38)(36,41,43,44)(39,45,48,46)(42,49,47,50);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,38,40,37)(36,41,43,44)(39,46,48,45)(42,49,47,50);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,22,24,20)(21,25,29,27)(23,30,31,26)(28,32,34,33)(35,36)(37,44)(38,41)(39,42)(40,43)(45,50)(46,49)(47,48);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,39,40,48)(36,42,43,47)(37,45,38,46)(41,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010018);