Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25472 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,11,14)(9,13,10,12)(15,16,19,17,20,22,18,21);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(15,17,18,16,20,21,19,22);(4,6)(15,16,20,22)(17,19,21,18);(7,10,11,9)(8,13,14,12)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353338);