Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52791 |
| 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 | |
4230 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);(3,4,7,8)(5,9,11,12)(6,14,15,10)(13,18,17,16)(19,20)(21,25)(22,24)(23,30)(26,31)(27,33)(28,29)(32,34)(38,46)(41,48)(44,50)(47,49);(4,8)(5,11)(10,14)(13,17)(20,22)(25,29)(26,30)(32,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,18)(15,17)(19,22,24,20)(21,25,28,29)(23,30,31,26)(27,32,34,33)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,6,7,15)(4,10,8,14)(5,13,11,17)(9,16,12,18)(19,23,24,31)(20,26,22,30)(21,27,28,34)(25,32,29,33)(35,38)(36,41)(37,44)(39,46)(40,47)(42,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999727);