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×#54642 |
| 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 | |
4278 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)(12,17)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34)(35,36,40,37)(38,41,45,43)(39,44,47,42)(46,49,50,48);(4,8)(9,14)(10,15)(16,18)(20,21)(22,29)(26,28)(30,34)(36,37)(38,45)(39,47)(48,49);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32)(35,37,40,36)(38,41,45,43)(39,44,47,42)(46,48,50,49);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,38,40,45)(36,41,37,43)(39,46,47,50)(42,48,44,49);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)(35,39)(36,42)(37,44)(38,46)(40,47)(41,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007010);