Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1444 |
| prime radical | C24 | C4⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(3,7)(5,8)(9,10,14,18,15,19,13,30)(11,20,23,33,24,32,22,16)(12,25,28,35,29,39,27,17)(21,31,26,36,38,40,37,34);(2,5,6,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,33,30,32)(12,26,27,21,29,37,28,38)(17,34,25,31,35,36,39,40);(1,2)(3,8)(4,6)(5,7)(9,12,13,27,15,29,14,28)(10,17,18,25,19,35,30,39)(11,21,23,26,24,38,22,37)(16,31,32,34,33,40,20,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fh,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28682);