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×#55875 |
| 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 | |
4354 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)(5,9)(6,8)(7,10)(11,12,15,16)(13,19,21,17)(14,22,23,18)(20,24,26,25)(27,28,32,35)(29,30,37,33)(31,34,39,41)(36,38,42,40);(3,5,8,7)(4,9,6,10)(12,16)(14,23)(17,19)(20,26)(31,39)(34,41)(36,42)(38,40);(3,6,8,4)(5,9,7,10)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25)(27,29,32,37)(28,33,35,30)(31,36,39,42)(34,40,41,38);(3,7,8,5)(4,9,6,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,30,32,33)(28,29,35,37)(31,38,39,40)(34,36,41,42);(11,14,15,23)(12,18,16,22)(13,20,21,26)(17,24,19,25)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010247);