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×#58789 |
| 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 | |
4262 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)(5,8)(6,11)(7,12)(9,13)(10,15)(14,16)(17,18)(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32)(35,36,40,43)(37,41,38,45)(39,42,47,50)(44,48,46,49);(4,5)(6,13)(10,12)(14,18)(20,22)(25,27)(26,30)(32,33)(35,37,40,38)(36,41,43,45)(39,46,47,44)(42,49,50,48);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,38,40,37)(36,41,43,45)(39,46,47,44)(42,48,50,49);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,49)(47,50);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23,24,31)(20,26,22,30)(21,28,29,34)(25,32,27,33)(35,39,40,47)(36,42,43,50)(37,44,38,46)(41,48,45,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012143);