Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C25 | M16 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,10,16,12,15,13,18,11,17,20,14,9,7,8,19);(5,7,10,19,9,17,8,14,11,15,20,18,12,6,13,16);(1,2,3,4)(5,8,9,20,11,13,12,10)(6,14,15,19,17,16,7,18)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47872);