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×#54457 |
| 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 | |
4170 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)(5,11)(6,8)(7,14)(9,12)(10,15)(13,16)(17,18)(19,20,24,25)(21,23,29,26)(22,30,31,27)(28,34,32,33)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,50)(47,49);(4,6)(5,12)(10,14)(13,18)(20,25)(22,31)(23,26)(28,32)(35,37,40,45)(36,41,43,38)(39,44,47,50)(42,48,49,46);(3,5)(4,9)(6,11)(7,13)(8,12)(10,16)(14,17)(15,18)(19,21)(20,26)(22,32)(23,25)(24,29)(27,33)(28,31)(30,34)(35,37)(36,38)(39,44)(40,45)(41,43)(42,46)(47,50)(48,49);(3,6,8,4)(5,11,12,9)(7,14,15,10)(13,17,18,16)(19,22,24,31)(20,27,25,30)(21,28,29,32)(23,33,26,34)(35,38)(36,37)(39,46)(40,41)(42,44)(43,45)(47,48)(49,50);(3,7,8,15)(4,10,6,14)(5,13,12,18)(9,16,11,17)(19,23,24,26)(20,21,25,29)(22,33,31,34)(27,28,30,32)(35,39,40,47)(36,42,43,49)(37,44,45,50)(38,46,41,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010322);