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×#50991 |
| 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 | |
4366 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);(3,4,7,10)(5,8,6,9)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,24)(25,26)(27,28,31,32)(29,35,36,33)(30,34,39,40)(37,42,38,41);(3,5,7,6)(4,8,10,9)(12,16)(17,19)(18,22)(24,25)(28,32)(29,36)(34,40)(37,38);(3,4)(5,8)(6,9)(7,10)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25)(27,29,31,36)(28,33,32,35)(30,38,39,37)(34,42,40,41);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,30,31,39)(28,34,32,40)(29,37,36,38)(33,41,35,42);(3,6,7,5)(4,9,10,8)(11,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,25)(21,26)(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, 7613825);