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 | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(6,8)(9,10,15,20,17,21,16,29)(11,22,26,35,28,38,27,39)(12,13,19,31,30,32,33,14)(18,34,36,40,37,25,23,24);(1,2,3,4)(9,11,14,25,16,27,32,40,17,28,31,34,15,26,13,24)(10,18,19,35,20,36,30,38,21,37,33,39,29,23,12,22);(5,6,7,8)(9,12)(10,13)(11,23)(14,29)(15,19)(16,33)(17,30)(18,26)(20,31)(21,32)(22,24)(25,39)(27,37)(28,36)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^c=dh,f^c=fhi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56448);