Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41740 |
| 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 | |
4562 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);(5,9)(8,10)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,28)(29,31)(30,38)(32,33)(34,37)(35,41)(36,42)(39,40);(3,4,7,6)(5,8,9,10)(12,16)(17,19)(18,23)(24,25)(28,33)(31,34)(35,38)(40,42);(3,5,7,9)(4,8,6,10)(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,29)(28,34)(30,39)(31,33)(32,37)(35,40)(36,41)(38,42);(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30,32,41)(28,35,33,38)(29,36,37,39)(31,40,34,42);(3,6,7,4)(5,10,9,8)(11,14,15,22)(12,18,16,23)(13,20,21,26)(17,24,19,25)(27,31,32,34)(28,29,33,37)(30,40,41,42)(35,36,38,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547052);