Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,6,10,7,11,5,13)(3,12,9,8,15,16,14,4)(18,24)(19,28)(20,31)(21,22)(25,30)(26,29)(27,32);(1,3,6,9,7,15,5,14)(2,8,10,16,11,4,13,12)(17,18)(19,29)(20,30)(21,26)(22,25)(23,27)(24,28)(31,32);(1,4)(2,9)(3,13)(5,16)(6,12)(7,8)(10,15)(11,14)(17,18)(19,30)(20,29)(21,26)(22,25)(23,27)(24,31)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=gi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43504);