Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | #532 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 300 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,3)(4,5)(7,8)(9,10,14,20,16,19,30,35,15,11,17,23,13,22,29,38)(12,24,27,39,28,21,33,37,25,36,31,40,26,18,32,34);(1,2,4,3,6,8,5,7)(9,11,14,20,16,22,30,35,15,10,17,23,13,19,29,38)(12,24,31,39,28,21,32,37,25,36,27,40,26,18,33,34);(1,3,4,8,6,7,5,2)(9,12,16,28,15,25,13,26)(10,18,22,36,11,21,19,24)(14,27,30,33,17,31,29,32)(20,34,38,40,23,37,35,39)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,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, 47989);