Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,10,8,9,11,12)(13,14,16,15,17,18,20,19);(1,2,3,4)(6,9)(7,11);(6,9)(10,12)(13,15,16,18,17,19,20,14)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=i,g^2=hi,b^a=bd,c^a=ci,d^a=dh,e^b=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12503);