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×#54093 |
| 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 | |
4186 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);(3,4,8,6)(5,9,13,11)(7,14,15,10)(12,17,18,16)(19,20)(21,26)(22,27)(23,30)(24,25)(28,33)(29,32)(31,34)(35,36,40,43)(37,41,38,45)(39,42,47,50)(44,48,46,49);(4,6)(9,11)(10,14)(16,17)(20,25)(21,29)(22,31)(23,33)(35,37,40,38)(36,41,43,45)(39,46,47,44)(42,49,50,48);(3,5,8,13)(4,9,6,11)(7,12,15,18)(10,16,14,17)(19,21,24,29)(20,26,25,32)(22,28,31,30)(23,27,33,34)(35,38,40,37)(36,41,43,45)(39,46,47,44)(42,48,50,49);(3,6,8,4)(5,11,13,9)(7,14,15,10)(12,17,18,16)(19,22,24,31)(20,27,25,34)(21,30,29,28)(23,32,33,26)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,49)(47,50);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23)(20,28)(21,27)(22,32)(24,33)(25,30)(26,31)(29,34)(35,39)(36,42)(37,44)(38,46)(40,47)(41,48)(43,50)(45,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006898);