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×#51304 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(11,12)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26)(27,28,31,32)(29,35,36,33)(30,34,39,40)(37,42,38,41);(3,4,7,6)(5,8,9,10)(12,17)(14,18)(20,23)(22,25)(28,32)(29,36)(34,40)(37,38);(3,5)(4,8)(6,10)(7,9)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22)(27,29,31,36)(28,33,32,35)(30,38,39,37)(34,42,40,41);(3,4,7,6)(5,8,9,10)(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,30,31,39)(28,34,32,40)(29,37,36,38)(33,41,35,42);(3,6,7,4)(5,10,9,8)(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,41)(38,42)(39,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614014);