Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| 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 | |
236 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 236 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)(3,5)(4,6)(7,8)(9,10,15,22,17,23,16,36)(11,24,27,40,29,19,28,39)(12,30,25,18,32,37,26,38)(13,33,34,21,14,35,31,20);(2,5)(6,7)(9,11,13,25,15,27,34,32,17,29,14,26,16,28,31,12)(10,18,21,19,36,30,33,40,23,38,20,24,22,37,35,39);(2,6)(4,8)(5,7)(9,12,31,28,16,26,14,29,17,32,34,27,15,25,13,11)(10,19,33,38,22,39,21,30,23,24,35,18,36,40,20,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=dh,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,e^c=eghi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47455);