Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(5,11)(9,14)(12,17)(16,18)(19,20,23,24)(21,27,28,25)(22,30,31,26)(29,32,34,33)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(4,8)(6,13)(9,14)(12,17)(20,24)(21,28)(26,30)(29,34)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21,23,28)(20,25,24,27)(22,29,31,34)(26,32,30,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,22,23,31)(20,26,24,30)(21,29,28,34)(25,32,27,33)(35,40)(36,43)(37,45)(38,46)(39,47)(41,48)(42,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013129);