Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1408 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1848 | C22 |
| socle | C2 | #7370 |
| prime radical | C23 | #356 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1848 ◁ G | |
| LECS | G ▷ #1848 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,6,11,20,28,7,13,18,29,5,12,21,30)(3,9,15,26,17,8,22,23,14,24,31,32,16,25,19,27);(2,8,12,9,13,25,11,24)(3,14)(4,18)(10,26,28,23,29,32,30,27)(16,17)(20,21)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=ghi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=degi,d^c=dgh,e^d=ehi,g^d=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1784);