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×#57937 |
| 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 | |
4230 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)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(4,6)(9,11)(10,14)(16,17)(23,31)(26,33)(28,34)(30,32)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,5,8,13)(4,9,6,11)(7,12,15,18)(10,16,14,17)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,6,8,4)(5,11,13,9)(7,14,15,10)(12,17,18,16)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34)(35,37,39,45)(36,40,42,43)(38,44,47,50)(41,48,46,49);(3,7,8,15)(4,10,6,14)(5,12,13,18)(9,16,11,17)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)(35,38,39,47)(36,41,42,46)(37,44,45,50)(40,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008618);