Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| 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 | |
409 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 409 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:
(1,2,7,13,6,14,27,11,8,12,29,31,26,15,5,24)(3,16,19,4,20,23,32,22,18,21,30,9,10,25,17,28);(1,3,6,10,8,18,26,20)(2,9,14,22,12,4,15,28)(5,19,7,17,27,30,29,32)(11,23,31,16,24,25,13,21);(1,4)(2,10)(3,15)(5,25)(6,28)(7,21)(8,9)(11,17)(12,20)(13,19)(14,18)(16,29)(22,26)(23,27)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,e^2=gh,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=eg,f^b=fgi,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214424, #72214645, #72214702, … |
| maximal normal subgroup of | #72214424, #72214645, #72214702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44781);