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×#57906 |
| 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);(4,8)(5,11)(10,15)(13,17)(19,20,23,24)(21,25,27,31)(22,26,29,32)(28,33,30,34)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(6,14)(10,15)(13,17)(16,18)(20,24)(21,27)(22,29)(33,34)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21)(20,25)(22,30)(23,27)(24,31)(26,34)(28,29)(32,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22)(20,26)(21,28)(23,29)(24,32)(25,33)(27,30)(31,34)(35,37,39,45)(36,40,42,43)(38,44,46,50)(41,47,48,49);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,32)(30,34)(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, 8008937);