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×#58556 |
| 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)(5,10)(6,9)(7,8)(12,16)(13,19)(18,23)(21,25)(27,28,33,36)(29,34,30,31)(32,40,41,35)(37,39,38,42);(3,5)(4,9)(6,8)(7,10)(14,22)(18,23)(21,25)(24,26)(27,29,33,30)(28,31,36,34)(32,38,41,37)(35,42,40,39);(5,6)(9,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,6,8,5)(4,9,7,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,31,33,34)(28,30,36,29)(32,39,41,42)(35,38,40,37);(3,7,8,4)(5,9,6,10)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,41)(34,42)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011378);