Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C211×C42 |
| direct product | C2×C4×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4,5,6);(7,8,11,14)(9,16,17,13)(10,18,20,22)(12,15,21,19);(7,9,11,17)(8,13,14,16)(10,19,20,15)(12,18,21,22);(7,10)(8,9)(11,20)(12,13)(14,17)(15,18)(16,21)(19,22);(7,9)(8,10)(11,17)(12,18)(13,15)(14,20)(16,19)(21,22)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533130);