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×#46223 |
| 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)(11,12,16,19)(13,20,17,14)(15,18,23,25)(21,26,24,22)(28,32)(29,35)(30,37)(40,42);(13,17)(14,20)(15,23)(18,25)(27,28,31,32)(29,33,35,39)(30,38,37,34)(36,42,41,40);(3,4,7,8)(5,9,6,10)(11,13,16,17)(12,14,19,20)(15,21,23,24)(18,22,25,26)(27,28)(29,33)(30,34)(31,32)(35,39)(36,40)(37,38)(41,42);(3,5,7,6)(4,9,8,10)(11,14)(12,17)(13,19)(15,22)(16,20)(18,24)(21,25)(23,26)(27,29,31,35)(28,33,32,39)(30,36,37,41)(34,40,38,42);(3,6,7,5)(4,10,8,9)(11,15,16,23)(12,18,19,25)(13,21,17,24)(14,22,20,26)(27,30,31,37)(28,34,32,38)(29,36,35,41)(33,40,39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609972);