Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×C4⋊#564 |
| 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 | |
6286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8)(9,15)(10,18)(11,14)(12,17)(13,19)(16,20)(21,22)(23,24,28,26)(25,31,33,29)(27,34,35,30)(32,36,38,37);(7,8,11,14)(9,12,17,15)(10,18,19,13)(16,21,22,20)(23,25,28,33)(24,29,26,31)(27,32,35,38)(30,36,34,37);(3,4,5,6)(7,8)(9,12)(10,13)(11,14)(15,17)(16,20)(18,19)(21,22);(7,9)(8,12)(10,16)(11,17)(13,20)(14,15)(18,21)(19,22)(23,26,28,24)(25,29,33,31)(27,34,35,30)(32,36,38,37);(7,10,11,19)(8,13,14,18)(9,16,17,22)(12,20,15,21)(23,27,28,35)(24,30,26,34)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610020);