Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C4×#3487 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3487 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4);(6,8)(10,11)(13,21)(15,17)(19,24);(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22);(5,6)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532858);