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×#50931 |
| 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);(3,4,7,5)(6,9,10,8)(11,12,16,17)(13,19,21,14)(15,23,24,18)(20,22,26,25)(27,28)(29,35)(30,34)(31,32)(33,36)(37,42)(38,41)(39,40);(3,5,7,4)(6,9,10,8)(12,17)(14,19)(18,23)(22,25)(28,31)(29,36)(34,39)(37,38);(3,6,7,10)(4,8,5,9)(11,13)(12,14)(15,20)(16,21)(17,19)(18,22)(23,25)(24,26)(27,29)(28,33)(30,38)(31,35)(32,36)(34,42)(37,40)(39,41);(11,14)(12,13)(15,22)(16,19)(17,21)(18,20)(23,26)(24,25)(27,30,32,40)(28,34,31,39)(29,37,36,38)(33,41,35,42);(11,15,16,24)(12,18,17,23)(13,20,21,26)(14,22,19,25)(27,31,32,28)(29,35,36,33)(30,39,40,34)(37,42,38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613886);