Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10)(11,15)(12,14)(13,20)(16,19)(17,23)(18,22)(21,24);(10,14)(11,17,18,21)(15,23,22,24)(16,20);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,21)(14,22)(16,23)(17,19)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=dh,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,d^c=dh,e^c=ei,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23241);