Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9064 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,6,14,26,12,9,15,25,31,8,28,5,24)(3,16,18,4,11,22,30,21,19,20,32,10,27,29,17,23);(1,3,8,11,9,19,6,27)(2,10,28,23,15,4,14,21)(5,18,25,30,26,32,7,17)(12,22,13,20,24,29,31,16);(1,4)(2,11)(3,14)(5,22)(6,23)(7,16)(8,21)(9,10)(12,17)(13,18)(15,27)(19,28)(20,25)(24,30)(26,29)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,e^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214383, #72214702, #72214756, … |
| maximal normal subgroup of | #72214383, #72214702, #72214756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44784);