Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,10,7,9,5,3)(4,12,14,16,15,11,13,8)(17,18,21,26,23,25,36,38,22,27,35,41,20,34,29,44)(19,28,31,42,33,40,46,47,32,43,45,48,30,24,37,39);(1,3,6,9,7,10,5,2)(4,11,14,16,15,12,13,8)(17,19,29,37,20,30,35,45,22,32,36,46,23,33,21,31)(18,24,38,42,34,43,26,39,27,40,44,48,25,28,41,47);(1,4,5,13,7,15,6,14)(2,8,10,12,9,16,3,11)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dgi,f^b=fgi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47997);