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×#56147 |
| 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 | |
4148 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)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(6,14)(9,16)(12,18)(15,17)(20,24)(22,31)(25,27)(28,34)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)(35,39)(36,42)(37,44)(38,45)(40,48)(41,47)(43,50)(46,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010252);