Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C22×#2044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
7294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(6,7)(9,10)(11,13)(12,14)(15,16);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16);(5,7,8,6)(13,16)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#24404, C22×#24260, #72283077, … |
| maximal normal subgroup of | C22×#24404, C22×#24260, #72283077, … |
| maximal quotient of | C22×#10995, C22×#13424, #72286046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339109);