Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19189 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,7,5,8,4,6)(10,16,20,26)(11,22)(12,25)(13,14)(17,27)(18,33,35,40)(19,31,21,30)(23,38)(24,28)(29,36,39,34);(2,6)(3,4)(7,8)(9,10,17,31,13,19,27,16,15,20,25,30,14,21,12,26)(11,18,32,33,28,36,38,29,24,35,37,40,22,34,23,39);(2,7)(3,4)(6,8)(9,11)(10,18)(12,23)(13,28)(14,22)(15,24)(16,29)(17,32)(19,36)(20,35)(21,34)(25,37)(26,39)(27,38)(30,40)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=eh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45852);