Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4566 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);(3,4,7,5)(6,9,10,8)(11,12,16,19)(13,17,20,14)(15,18,23,25)(21,24,26,22)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,42)(39,40);(3,5,7,4)(6,9,10,8)(13,20)(14,17)(15,23)(18,25)(28,31)(29,35)(34,39)(36,38);(3,6)(4,8)(5,9)(7,10)(11,13,16,20)(12,17,19,14)(15,21,23,26)(18,24,25,22)(27,29,32,35)(28,33,31,37)(30,38,40,36)(34,42,39,41);(11,14,16,17)(12,13,19,20)(15,22,23,24)(18,21,25,26)(27,30,32,40)(28,34,31,39)(29,36,35,38)(33,41,37,42);(11,15)(12,18)(13,21)(14,22)(16,23)(17,24)(19,25)(20,26)(27,31,32,28)(29,37,35,33)(30,39,40,34)(36,42,38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614224);