Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133834 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 90 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)(11,15)(12,22)(13,25,27,17)(14,18)(16,24,31,38)(19,33)(20,32,26,29)(21,30)(23,37)(28,35,40,39)(34,36);(2,5)(3,7)(9,11,14,21)(10,15,18,30)(12,19,23,34)(13,26,27,20)(16,28,31,40)(17,32,25,29)(22,36,37,33)(24,39,38,35);(9,12)(10,16)(11,19)(13,24)(14,23)(15,28)(17,22)(18,31)(20,35)(21,34)(25,37)(26,39)(27,38)(29,33)(30,40)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,d^c=di,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72227716, #72227722, #72227723, … |
| maximal normal subgroup of | #72227716, #72227722, #72227723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41242);