Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
694 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)(5,6,10,11,8,7,9,12)(13,14,17,20,16,19,18,15);(1,2,4,3)(6,7)(11,12)(13,15,18,19,16,20,17,14);(1,3,4,2)(5,7,8,6)(9,11,10,12)(14,19)(15,20)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,c^b=ci,e^c=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15625);