Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,6,8,7)(9,10,16,23,18,24,17,39)(11,25,29,33,30,31,40,32)(12,27,34,26,19,28,35,38)(13,22,14,37,36,21,15,20);(1,2)(3,4)(5,7,8,6)(9,11,18,30)(10,19,24,12)(13,28,36,27)(14,38,15,26)(16,29,17,40)(20,32,37,33)(21,31,22,25)(23,35,39,34);(2,4)(6,7)(9,12,13,31,16,34,14,32,18,19,36,25,17,35,15,33)(10,11,20,26,39,40,21,27,24,30,37,38,23,29,22,28)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56321);