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×#56219 |
| 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 | |
4316 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)(5,11,12,9)(6,14,15,10)(13,16,18,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(4,8)(5,12)(10,14)(13,18)(20,24)(21,27)(22,31)(32,34)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,5,7,12)(4,9,8,11)(6,13,15,18)(10,16,14,17)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,6,7,15)(4,10,8,14)(5,13,12,18)(9,16,11,17)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)(35,39)(36,42)(37,44)(38,45)(40,48)(41,47)(43,50)(46,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010182);