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×#58229 |
| 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 | |
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);(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(28,32)(29,35)(30,38)(40,42);(3,4)(5,6)(7,8)(9,10)(13,20)(16,19)(21,26)(23,25)(27,28,31,32)(29,36,35,33)(30,34,38,39)(37,42,41,40);(4,8)(5,9)(11,13,15,20)(12,16,18,19)(14,21,22,26)(17,23,24,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,5,7,9)(4,6,8,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,29,31,35)(28,33,32,36)(30,37,38,41)(34,40,39,42);(3,6)(4,9)(5,8)(7,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,30,31,38)(28,34,32,39)(29,37,35,41)(33,40,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011770);