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×#58218 |
| 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 | |
4598 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);(12,16)(13,19)(18,23)(20,25)(27,28,32,36)(29,34,30,33)(31,39,40,35)(37,41,38,42);(3,4)(5,6)(7,8)(9,10)(12,16)(17,22)(18,23)(24,26)(27,29,32,30)(28,33,36,34)(31,38,40,37)(35,42,39,41);(4,8)(5,9)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(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)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,28)(29,34)(30,33)(31,35)(32,36)(37,42)(38,41)(39,40);(3,6)(4,9)(5,8)(7,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(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, 8011754);