Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,4,3)(6,11,13,19)(7,15)(8,9)(12,16,14,20)(17,18);(5,6)(7,16,17,20)(8,12)(9,14)(10,13)(11,18,19,15);(1,3,4,2)(6,12,13,14)(7,17)(11,16,19,20)(15,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72281512, #72313480, #72313510, … |
| maximal normal subgroup of | #72281512, #72313480, #72313510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36000);