Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8)(9,10)(11,19)(12,13)(14,17)(15,20)(16,21)(18,22)(23,24,28,26)(25,31,33,29)(27,34,36,30)(32,37,35,38);(8,13)(10,14)(15,19)(18,21)(23,25,28,33)(24,29,26,31)(27,35,36,32)(30,38,34,37);(3,4,5,6)(7,9)(8,14)(10,13)(11,16)(12,17)(15,21)(18,19)(20,22)(23,25)(24,29)(26,31)(27,32)(28,33)(30,37)(34,38)(35,36);(7,10,12,14)(8,9,13,17)(11,18,20,21)(15,16,19,22)(23,26,28,24)(25,29,33,31)(27,34,36,30)(32,37,35,38);(7,11,12,20)(8,15,13,19)(9,16,17,22)(10,18,14,21)(23,27,28,36)(24,30,26,34)(25,32,33,35)(29,37,31,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608128);