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×#56175 |
| 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);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(4,5)(6,13)(7,15)(16,17)(20,25)(22,27)(23,32)(28,34)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,6,8,13)(4,9,5,11)(7,14,15,18)(10,16,12,17)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23,24,32)(20,26,25,31)(21,28,29,34)(22,30,27,33)(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, 8010275);