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×#11137 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2670 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)(7,8,12,16,13,17,11,26)(9,18,22,29,21,34,20,14)(10,15,24,32,25,31,27,30)(19,33,35,38,23,37,36,28);(7,9,11,20,13,21,12,22)(8,14,16,18,17,29,26,34)(10,23,24,36,25,19,27,35)(15,28,30,37,31,38,32,33);(4,6)(7,10)(8,15)(9,19)(11,27)(12,24)(13,25)(14,28)(16,32)(17,31)(18,33)(20,36)(21,23)(22,35)(26,30)(29,38)(34,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265026);