Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97642 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,10,8,7,11,12)(13,14,16,18,17,20,15,19);(1,2)(3,4)(9,11)(10,12)(14,18)(15,16)(19,20);(2,4)(5,7)(6,8)(9,10)(11,12)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=eh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18909);