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×#55905 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,8,6)(5,9,13,11)(7,14,15,10)(12,17,18,16)(19,20)(21,26)(22,28)(23,25)(24,27)(29,33)(30,34)(31,32)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(4,6)(9,11)(10,14)(16,17)(19,21,25,22)(20,26,23,28)(24,30,32,29)(27,34,31,33)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,5,8,13)(4,9,6,11)(7,12,15,18)(10,16,14,17)(19,22,25,21)(20,26,23,28)(24,30,32,29)(27,33,31,34)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,6,8,4)(5,11,13,9)(7,14,15,10)(12,17,18,16)(19,23,25,20)(21,28,22,26)(24,31,32,27)(29,34,30,33)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,7,8,15)(4,10,6,14)(5,12,13,18)(9,16,11,17)(19,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,33)(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, 8010336);