Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11194 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2546 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);(7,8)(9,19)(10,15,26,34)(11,18)(12,16)(13,17)(14,21)(20,31,37,35)(22,29)(23,30)(24,28,36,38)(25,33,27,32);(3,4)(5,6)(7,9)(8,14)(10,24)(11,23)(12,21)(13,22)(15,31)(16,29)(17,30)(18,19)(20,27)(25,37)(26,36)(28,33)(32,38)(34,35);(4,6)(7,10,11,25,13,26,12,27)(8,15,16,32,17,34,18,33)(9,20,21,36,22,37,23,24)(14,28,19,35,30,38,29,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265164);