Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37193 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6918 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)(9,15)(10,17)(11,19)(12,13)(14,20)(16,18)(21,22)(26,34)(29,36)(32,38)(35,37);(8,13)(10,15)(14,19)(18,21)(23,24,27,30)(25,31,33,28)(26,29,34,36)(32,37,38,35);(3,4,5,6)(7,9)(8,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,22)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,22)(20,21)(23,25,27,33)(24,28,30,31)(26,32,34,38)(29,35,36,37);(7,11,12,20)(8,14,13,19)(9,16,17,22)(10,18,15,21)(23,26)(24,29)(25,32)(27,34)(28,35)(30,36)(31,37)(33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541799);