Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4676 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);(3,4,7,6)(5,9,10,8)(11,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,25)(22,26)(27,28,32,30)(29,35,37,33)(31,38,40,34)(36,41,39,42);(12,17)(14,18)(19,23)(22,25)(27,29,32,37)(28,33,30,35)(31,39,40,36)(34,42,38,41);(3,5,7,10)(4,8,6,9)(11,13)(12,18)(14,17)(15,20)(16,21)(19,25)(22,23)(24,26)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,42)(39,40);(3,6,7,4)(5,9,10,8)(11,14,16,18)(12,13,17,21)(15,22,24,25)(19,20,23,26)(27,30,32,28)(29,33,37,35)(31,38,40,34)(36,41,39,42);(11,15,16,24)(12,19,17,23)(13,20,21,26)(14,22,18,25)(27,31,32,40)(28,34,30,38)(29,36,37,39)(33,41,35,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608593);