Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23);(2,6)(3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(2,6)(7,8);(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=hi,g^2=i,b^a=bf,d^a=di,f^a=fh,c^b=ci,g^b=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249840);