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×#57753 |
| 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 | |
4180 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)(13,17)(19,20)(21,27)(22,24)(23,30)(25,29)(26,31)(28,32)(33,34)(35,36,40,43)(37,38,45,41)(39,47,48,42)(44,49,50,46);(6,14)(10,15)(13,17)(16,18)(20,22)(23,31)(25,27)(28,34)(35,37,40,45)(36,41,43,38)(39,44,48,50)(42,49,47,46);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,37)(36,41)(38,43)(39,44)(40,45)(42,49)(46,47)(48,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,17)(15,18)(19,22,24,20)(21,25,29,27)(23,30,31,26)(28,32,34,33)(35,38,40,41)(36,37,43,45)(39,46,48,49)(42,44,47,50);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,39)(36,42)(37,44)(38,46)(40,48)(41,49)(43,47)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008757);