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×#46146 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8,12,9)(10,15,18,13)(11,16,19,14)(17,20,22,21)(23,24)(25,31)(26,34)(27,28)(29,33)(30,35)(32,36)(37,38);(8,9)(11,19)(13,15)(17,22)(24,28)(29,31)(30,34)(36,37);(3,4,5,6)(7,9,12,8)(10,15,18,13)(11,16,19,14)(17,21,22,20)(23,25)(24,29)(26,32)(27,33)(28,31)(30,36)(34,37)(35,38);(7,10,12,18)(8,13,9,15)(11,17,19,22)(14,20,16,21)(23,24)(25,29)(26,30)(27,28)(31,33)(32,36)(34,35)(37,38);(7,11,12,19)(8,14,9,16)(10,17,18,22)(13,20,15,21)(23,26,27,35)(24,30,28,34)(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, 7609984);