Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
468 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(5,6,7,12,8,13,15,20,9,10,14,17,16,11,18,19);(1,2,3,4)(6,10)(11,13);(6,11,10,13)(7,14)(12,20,17,19)(15,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=dg,g^a=gi,e^b=eg,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239962, #72250602, #72250615, … |
| maximal normal subgroup of | #72239962, #72250602, #72250615, … |
| maximal quotient of | #72240003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34179);