Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,12,16,14)(13,15)(18,22)(19,23)(20,24);(2,4)(5,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,19)(20,23,24,22);(9,12,13,14)(10,11,15,16)(17,19,21,18)(20,22,24,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15183);