Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142308 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,4)(6,12)(7,16)(8,18)(9,10)(13,15)(14,20)(17,19);(5,6)(7,17)(8,13)(9,19)(10,15)(11,14)(12,16)(18,20);(2,4)(6,13)(7,18)(8,16)(9,10)(12,19)(14,17)(15,20)Polycyclic presentation:
2^9|d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250550, #72250597, #72250602, … |
| maximal normal subgroup of | #72250550, #72250597, #72250602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50009);