Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,11,9,13,16,19,10,14,17,20,8,18,15,12);(2,4)(6,11,13,19,14,20,18,12);(5,7,8,15,10,17,9,16)(6,12,13,11,14,19,18,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,c^b=cgh,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213075, #72214259, #72214702, … |
| maximal normal subgroup of | #72214259, #72214702, #72214744, … |
| maximal quotient of | #72213035, #72213040, #72213041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34090);