Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C26×C4 |
| unambiguous semidirect product | C22⋊#1273 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C42 | C23 |
| commutator | C22 | C2×C82 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,13,12,14,15,16);(2,4)(17,18,19,21,20,22,23,24);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce224 conjugacy classes — view the full character table.
| maximal subgroup of | #72286292, #72286299, #72286354, … |
| maximal normal subgroup of | #72286292, #72286299, #72286354, … |
| maximal quotient of | #72286531, #72322486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12676);