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×#58680 |
| 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);(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(4,8)(6,14)(9,11)(12,18)(20,22)(25,27)(26,30)(32,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32)(35,37,39,45)(36,40,42,43)(38,44,46,50)(41,47,48,49);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,38,39,46)(36,41,42,48)(37,44,45,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012049);