Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45943 |
| 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 | |
4594 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);(4,8)(9,10)(12,16)(13,19)(18,23)(20,25)(27,28)(29,35)(30,38)(31,34)(32,37)(33,39)(36,40)(41,42);(12,16)(13,19)(14,21)(24,26)(27,28,31,34)(29,32,37,35)(30,38,39,33)(36,41,42,40);(3,4,7,8)(5,9,6,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,37)(36,40)(38,39)(41,42);(3,5,7,6)(4,9,8,10)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,29,31,37)(28,32,34,35)(30,36,39,42)(33,40,38,41);(3,6,7,5)(4,10,8,9)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,30)(28,33)(29,36)(31,39)(32,40)(34,38)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609641);