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×#58555 |
| 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);(3,4)(5,9)(6,7)(8,10)(12,16)(13,19)(18,23)(21,25)(27,28,32,35)(29,33,38,42)(30,31,34,41)(36,40,39,37);(4,7)(8,9)(14,22)(18,23)(21,25)(24,26)(27,29,32,38)(28,33,35,42)(30,36,34,39)(31,40,41,37);(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,33)(30,39)(31,37)(32,38)(34,36)(35,42)(40,41);(3,4)(5,8)(6,7)(9,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,34)(28,31,35,41)(29,36,38,39)(33,40,42,37);(3,5,6,10)(4,8,7,9)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31)(28,34)(29,37)(30,35)(32,41)(33,36)(38,40)(39,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011395);