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×#51003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4534 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)(13,20,21,17)(14,18,24,15)(22,26,25,23)(27,28,33,31)(29,36,30,34)(32,39,40,35)(37,41,38,42);(13,21)(14,24)(15,18)(17,20)(27,29,33,30)(28,34,31,36)(32,37,40,38)(35,41,39,42);(3,4)(5,9)(6,10)(7,8)(11,13)(12,17)(14,25)(15,23)(16,21)(18,26)(19,20)(22,24)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,5,7,6)(4,9,8,10)(11,14)(12,18)(13,22)(15,19)(16,24)(17,23)(20,26)(21,25)(27,31,33,28)(29,34,30,36)(32,39,40,35)(37,41,38,42);(3,6,7,5)(4,10,8,9)(11,15,16,18)(12,14,19,24)(13,23,21,26)(17,25,20,22)(27,32,33,40)(28,35,31,39)(29,37,30,38)(34,41,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613804);