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×#58191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4636 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);(11,12)(13,17)(14,21)(15,16)(18,22)(19,25)(20,23)(24,26)(28,29)(30,37)(31,39)(40,41);(3,4)(5,6)(7,8)(9,10)(12,16)(17,23)(18,21)(24,25)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(4,8)(5,9)(11,13,15,20)(12,17,16,23)(14,19,22,26)(18,24,21,25)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,5,7,9)(4,6,8,10)(11,12)(13,17)(14,18)(15,16)(19,24)(20,23)(21,22)(25,26)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,6)(4,9)(5,8)(7,10)(11,14)(12,18)(13,19)(15,22)(16,21)(17,24)(20,26)(23,25)(27,31,32,39)(28,34,29,36)(30,38,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011830);