Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(10,13)(11,15)(18,22,23,24);(1,3,6,2)(4,8,5,7)(9,10,11,14,12,13,15,16)(17,18,20,22,21,23,19,24);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bd,d^a=dg,f^a=fghi,d^b=dh,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3259);