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×#58186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28,32,36)(29,34,30,33)(31,39,40,35)(37,41,38,42);(3,4)(5,6)(7,8)(9,10)(14,22)(17,24)(20,26)(23,25)(27,29,32,30)(28,33,36,34)(31,38,40,37)(35,42,39,41);(4,8)(5,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,30,32,29)(28,34,36,33)(31,38,40,37)(35,42,39,41);(3,5,7,9)(4,6,8,10)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,28)(29,34)(30,33)(31,35)(32,36)(37,42)(38,41)(39,40);(3,6)(4,9)(5,8)(7,10)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(27,31)(28,35)(29,37)(30,38)(32,40)(33,41)(34,42)(36,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011756);