Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60325 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,17,14,13,16,18);(3,5)(4,7)(6,10)(8,9)(12,13)(17,18);(5,10)(7,9)(12,13)(17,18);(4,8)(7,9);(11,13,15,18,14,12,16,17)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481681);