Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1934 |
| prime radical | C23 | #465 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10,15,22,17,24,16,23)(11,25,29,40,12,31,30,37)(13,34,14,36,26,21,35,20)(18,28,38,32,39,33,19,27);(9,11,26,32,15,29,35,33,17,12,13,27,16,30,14,28)(10,18,20,25,23,19,21,37,24,39,36,31,22,38,34,40);(2,5)(3,8)(6,7)(9,12,13,32,15,30,14,33,17,11,26,27,16,29,35,28)(10,19,20,37,23,39,21,31,24,38,36,40,22,18,34,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=d,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=egh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47450);