Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52784 |
| 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 | |
4164 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);(3,4,7,8)(5,9,13,11)(6,14,15,10)(12,17,18,16)(22,30)(25,32)(28,34)(31,33)(35,36,40,43)(37,41,38,45)(39,47,48,42)(44,50,46,49);(4,8)(9,11)(10,14)(16,17)(19,20,23,26)(21,24,29,27)(22,25,30,32)(28,31,34,33)(35,37,40,38)(36,41,43,45)(39,44,48,46)(42,49,47,50);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,17)(15,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34)(35,36)(37,41)(38,45)(39,42)(40,43)(44,49)(46,50)(47,48);(3,4)(5,11)(6,10)(7,8)(9,13)(12,17)(14,15)(16,18)(19,21)(20,24)(22,28)(23,29)(25,31)(26,27)(30,34)(32,33)(35,38,40,37)(36,41,43,45)(39,46,48,44)(42,49,47,50);(3,6,7,15)(4,10,8,14)(5,12,13,18)(9,16,11,17)(19,22)(20,25)(21,28)(23,30)(24,31)(26,32)(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, 7999723);