Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51839 |
| 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 | |
4616 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)(11,12,16,19)(13,14,17,21)(15,18,23,25)(20,22,24,26)(27,28,32,33)(29,37,38,34)(30,35,36,31)(39,42,41,40);(3,5,7,6)(4,8,10,9)(13,17)(14,21)(15,23)(18,25)(28,33)(29,38)(31,35)(39,41);(3,4)(5,8)(6,9)(7,10)(11,13)(12,14)(15,20)(16,17)(18,22)(19,21)(23,24)(25,26)(27,29,32,38)(28,34,33,37)(30,41,36,39)(31,40,35,42);(3,6,7,5)(4,9,10,8)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,30,32,36)(28,35,33,31)(29,39,38,41)(34,40,37,42);(11,15,16,23)(12,18,19,25)(13,20,17,24)(14,22,21,26)(27,31,32,35)(28,36,33,30)(29,40,38,42)(34,39,37,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614973);