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×#56957 |
| 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)(5,9)(6,13)(7,8)(10,14)(11,17)(12,15)(16,18)(19,20,23,24)(21,25,27,31)(22,26,29,32)(28,33,30,34)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(4,8)(9,15)(10,13)(16,17)(20,24)(21,27)(22,29)(33,34)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,7,12)(4,9,8,15)(6,11,14,18)(10,16,13,17)(19,21)(20,25)(22,30)(23,27)(24,31)(26,34)(28,29)(32,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,4)(5,9)(6,10)(7,8)(11,16)(12,15)(13,14)(17,18)(19,22)(20,26)(21,28)(23,29)(24,32)(25,33)(27,30)(31,34)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,6)(4,10)(5,11)(7,14)(8,13)(9,16)(12,18)(15,17)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,32)(30,34)(35,38,39,46)(36,41,42,48)(37,44,45,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007355);