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×#54215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 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);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(4,5)(6,13)(7,15)(16,17)(20,25)(22,26)(28,31)(30,33)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,6,8,13)(4,9,5,11)(7,14,15,18)(10,16,12,17)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,33)(27,34)(35,39)(36,43)(37,45)(38,46)(40,48)(41,49)(42,50)(44,47)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008812);