Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37197 |
| 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 | |
4366 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);(3,4,6,8)(5,9,10,7)(11,12,15,16)(13,19,21,17)(14,22,23,18)(20,24,26,25)(30,38)(33,40)(36,42)(39,41);(12,16)(17,19)(18,22)(24,25)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,25)(23,26)(27,28)(29,32)(30,33)(31,34)(35,37)(36,39)(38,40)(41,42);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(11,14,15,23)(12,18,16,22)(13,20,21,26)(17,24,19,25)(27,30)(28,33)(29,36)(31,38)(32,39)(34,40)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541800);