Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1362 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,9)(8,11)(10,12)(13,14,15,18,16,17,19,20);(6,9,11,10);(1,2,3,4)(14,17)(18,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,b^a=bd,c^a=ci,d^a=dh,f^a=fh176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#7867, #72240066, #72240068, … |
| maximal normal subgroup of | C4×#7867, #72240066, #72240068, … |
| maximal quotient of | #72237306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12463);