Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | #3100 | C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,7)(3,8)(4,5)(9,10,14,21,16,20,32,31,15,19,22,36,13,11,17,24)(12,25,29,39,30,26,37,40,27,23,34,38,28,18,33,35);(1,2,5,3,6,8,4,7)(9,11,22,24,16,10,17,21,15,20,14,31,13,19,32,36)(12,26,29,35,30,23,37,39,27,18,34,40,28,25,33,38);(1,3,5,8,6,7,4,2)(9,12,13,28,15,27,16,30)(10,18,20,25,19,26,11,23)(14,29,17,33,22,34,32,37)(21,35,31,39,36,40,24,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cgh,d^b=di,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47986);