Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1434 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,8,4,7,6,3)(9,10)(11,20,24,35)(12,25)(13,18)(14,28)(15,19)(16,22,31,23)(17,26)(21,30,37,40)(27,32)(29,33)(34,38,39,36);(1,3,6,7,4,8,5,2)(9,11)(10,16,19,31)(12,21)(13,22)(14,23)(15,24)(17,30,33,40)(18,20,28,35)(25,39,32,34)(26,36)(27,37)(29,38);(2,7)(3,8)(9,12,13,26,15,27,14,29)(10,17,18,32,19,33,28,25)(11,21,22,36,24,37,23,38)(16,30,20,34,31,40,35,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gh,c^b=ci,f^b=fh,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15631);