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×#57798 |
| 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 | |
4148 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)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31)(36,37)(38,45)(39,47)(48,49);(4,8)(5,12)(10,14)(13,18)(22,30)(25,32)(28,34)(31,33)(35,36,40,37)(38,43,45,41)(39,44,47,42)(46,48,50,49);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33)(35,37,40,36)(38,41,45,43)(39,44,47,42)(46,48,50,49);(3,5,7,12)(4,9,8,11)(6,13,15,18)(10,16,14,17)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34)(35,38,40,45)(36,41,37,43)(39,46,47,50)(42,48,44,49);(3,6,7,15)(4,10,8,14)(5,13,12,18)(9,16,11,17)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)(35,39,40,47)(36,42,37,44)(38,46,45,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008922);