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×#56428 |
| 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 | |
4598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(11,12,16,17)(13,14,18,21)(15,23,24,19)(20,26,25,22)(28,32)(29,35)(30,38)(40,42);(3,4)(5,9)(6,8)(7,10)(12,17)(14,21)(19,23)(22,26)(27,28,31,32)(29,36,35,33)(30,34,38,39)(37,42,41,40);(3,4,6,8)(5,7,10,9)(11,13,16,18)(12,14,17,21)(15,20,24,25)(19,22,23,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,4)(5,7)(6,8)(9,10)(11,14)(12,18)(13,17)(15,22)(16,21)(19,25)(20,23)(24,26)(27,29,31,35)(28,33,32,36)(30,37,38,41)(34,40,39,42);(3,5)(4,7)(6,10)(8,9)(11,15,16,24)(12,19,17,23)(13,20,18,25)(14,22,21,26)(27,30,31,38)(28,34,32,39)(29,37,35,41)(33,40,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006331);