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×#54736 |
| 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);(4,8)(5,11)(10,15)(12,17)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(4,8)(9,14)(10,15)(16,18)(23,31)(26,33)(28,34)(30,32)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)(35,39,40,48)(36,43,44,47)(37,45,38,46)(41,49,42,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010993);