Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8032 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,15)(10,18)(11,14)(12,17)(13,19)(16,21)(20,22)(23,24)(25,32)(26,30)(27,36)(28,31)(29,34)(33,38)(35,37);(7,9)(8,12,14,20)(10,17)(11,16)(13,15,18,21)(19,22)(23,25,27,35,28,34,26,33)(24,29,30,37,31,32,36,38);(9,16)(12,20)(15,21)(17,22)(23,26,28,27)(24,30,31,36)(25,33,34,35)(29,37,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88766);