Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1825 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8711 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,6,11,7,12,24,23,8,13,5,21,22,4,19,10)(3,14,17,29,18,20,32,25,15,9,16,26,30,27,31,28);(1,3,7,18,8,15,22,30)(2,9,4,20,13,14,12,27)(5,16,19,31,6,17,24,32)(10,25,21,29,23,28,11,26);(1,4)(2,7)(3,14,15,9)(5,10)(6,23)(8,12)(11,19)(13,22)(16,29,17,26)(18,20,30,27)(21,24)(25,32,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42487);