Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135322 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,22,24,16)(12,17)(13,21)(14,20)(15,18)(19,28)(23,36,39,30)(25,32,26,33)(27,35)(29,34)(31,40,37,38);(9,11)(10,16)(12,23)(13,25)(14,26)(15,24)(17,30)(18,22)(19,31)(20,32)(21,33)(27,38)(28,40)(29,39)(34,36)(35,37);(2,5)(3,7)(10,17,18,34)(12,27)(13,14)(16,30,22,36)(19,21,35,20)(23,38)(25,26)(28,29)(31,33,37,32)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,f^b=fi,e^c=eh,f^c=fgh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240840 |
| maximal normal subgroup of | #72240840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42611);