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×#50584 |
| 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 | |
4480 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)(11,12,16,20)(13,14,18,17)(15,19,23,26)(21,22,25,24)(27,28)(29,36)(30,33)(31,39)(32,35)(34,40)(37,41)(38,42);(13,18)(14,17)(15,23)(19,26)(27,29,32,30)(28,33,35,36)(31,37,40,38)(34,41,39,42);(3,4,7,8)(5,9,6,10)(11,13)(12,17)(14,20)(15,21)(16,18)(19,24)(22,26)(23,25)(27,30,32,29)(28,33,35,36)(31,38,40,37)(34,41,39,42);(3,5,7,6)(4,9,8,10)(11,14,16,17)(12,18,20,13)(15,22,23,24)(19,25,26,21)(27,28)(29,36)(30,33)(31,34)(32,35)(37,42)(38,41)(39,40);(3,6,7,5)(4,10,8,9)(11,15,16,23)(12,19,20,26)(13,21,18,25)(14,22,17,24)(27,31,32,40)(28,34,35,39)(29,37,30,38)(33,41,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613239);