Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46204 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(11,12,16,17)(13,14,18,21)(15,23,24,19)(20,26,25,22)(28,32)(29,35)(30,37)(40,42);(3,4,7,6)(5,8,9,10)(12,17)(14,21)(19,23)(22,26)(27,28,31,32)(29,33,35,39)(30,38,37,34)(36,42,41,40);(3,5,7,9)(4,8,6,10)(11,13,16,18)(12,14,17,21)(15,20,24,25)(19,22,23,26)(27,28)(29,33)(30,34)(31,32)(35,39)(36,40)(37,38)(41,42);(11,14)(12,18)(13,17)(15,22)(16,21)(19,25)(20,23)(24,26)(27,29,31,35)(28,33,32,39)(30,36,37,41)(34,40,38,42);(3,6,7,4)(5,10,9,8)(11,15)(12,19)(13,20)(14,22)(16,24)(17,23)(18,25)(21,26)(27,30,31,37)(28,34,32,38)(29,36,35,41)(33,40,39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609942);