Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97486 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,13)(11,15)(14,16)(17,18,21,23,22,19,20,24);(1,2)(3,7)(4,6)(5,8)(17,19,22,18)(20,23,21,24);(9,10,12,14)(11,16,15,13)(17,18,22,19)(20,24,21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5951);