Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4,9,14)(5,15,17,27)(6,19,21,28)(7,13,8,12)(10,22,25,18)(11,23,26,20)(16,24,30,34)(29,31,33,32)(35,36,40,43)(37,45,47,59)(38,42,50,57)(39,51,54,44)(41,52,56,48)(46,62,63,61)(49,60,53,58)(55,65,66,64);(3,5)(4,10)(6,16)(7,22)(8,18)(9,17)(11,24)(12,27)(13,15)(14,25)(19,33)(20,32)(21,30)(23,31)(26,34)(28,29)(35,37,40,47)(36,41,43,56)(38,46,50,63)(39,52,54,48)(42,55,57,66)(44,59,51,45)(49,65,53,64)(58,61,60,62);(3,6,9,21)(4,11,14,26)(5,16,17,30)(7,20,8,23)(10,24,25,34)(12,19,13,28)(15,29,27,33)(18,31,22,32)(35,38,40,50)(36,42,43,57)(37,46,47,63)(39,53,54,49)(41,55,56,66)(44,60,51,58)(45,61,59,62)(48,65,52,64);(5,17)(10,25)(15,27)(16,30)(18,22)(24,34)(29,33)(31,32)(36,43)(37,47)(42,57)(44,51)(46,63)(48,52)(58,60)(64,65)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258908);