Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C2×#8025 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
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)(25,31)(26,29)(27,30)(28,33)(32,37)(34,36)(35,38);(7,9)(8,12,14,20)(10,17)(11,16)(13,15,18,21)(19,22)(23,25)(24,28,30,31)(26,34)(27,33)(29,37,38,36)(32,35);(9,16)(12,20)(15,21)(17,22)(23,26,27,35)(24,29,30,38)(25,32,33,34)(28,36,31,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88718);