Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C22×D8 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 310 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18,20,24)(19,22,28,30)(21,29,27,25)(23,32,31,26);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(18,22)(19,26)(21,27)(23,25)(24,30)(28,32)(29,31);(18,22)(19,26)(21,27)(23,25)(24,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=gi,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43670);