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×#56198 |
| 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 | |
4246 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);(4,8)(5,11)(10,15)(13,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(6,14)(10,15)(13,17)(16,18)(20,24)(21,27)(22,31)(32,34)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)(35,39,40,48)(36,43,44,47)(37,45,38,46)(41,49,42,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007081);