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×#57064 |
| 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 | |
4148 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)(5,9)(6,14)(7,8)(10,15)(11,12)(13,18)(16,17)(19,20,24,25)(21,22,26,29)(23,31,32,27)(28,34,33,30)(36,40)(37,43)(38,46)(48,50);(4,8)(5,11)(6,15)(16,18)(20,25)(22,29)(27,31)(30,34)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,18)(15,17)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6,7,15)(4,10,8,14)(5,13,11,17)(9,16,12,18)(19,23,24,32)(20,27,25,31)(21,28,26,33)(22,30,29,34)(35,38)(36,42)(37,45)(39,46)(40,47)(41,48)(43,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006914);