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×#56748 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4734 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);(3,4,7,9)(5,10,6,8)(12,16)(13,19)(18,23)(21,25)(27,28)(29,34)(30,35)(31,39)(32,33)(36,40)(37,38)(41,42);(3,5)(4,8)(6,7)(9,10)(14,22)(18,23)(21,25)(24,26)(28,33)(30,34)(36,39)(38,41);(5,6)(8,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29,32,35)(28,34,33,30)(31,37,40,42)(36,41,39,38);(3,6,7,5)(4,8,9,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,34)(28,35,33,29)(31,38,40,41)(36,42,39,37);(3,4)(5,10)(6,8)(7,9)(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,41)(35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006716);