Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6,8,11,9,7,12,10)(13,14,16,19,18,20,17,15);(1,2,4,3)(13,15,17,20,18,19,16,14);(1,3,4,2)(5,7)(6,9)(8,10)(11,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,e^c=eh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15624);