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×#41809 |
| 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 | |
4600 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);(3,4,7,10)(5,8,6,9)(12,16)(17,19)(18,23)(24,25)(27,28)(29,30)(31,35)(32,33)(34,38)(36,37)(39,40)(41,42);(12,16)(14,22)(17,19)(20,26)(28,33)(29,34)(35,40)(36,41);(3,4)(5,8)(6,9)(7,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,29,32,34)(28,30,33,38)(31,36,39,41)(35,37,40,42);(3,5,7,6)(4,8,10,9)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30)(28,34)(29,33)(31,37)(32,38)(35,41)(36,40)(39,42);(3,6,7,5)(4,9,10,8)(11,14,15,22)(12,18,16,23)(13,20,21,26)(17,24,19,25)(27,31,32,39)(28,35,33,40)(29,36,34,41)(30,37,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547160);