Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130498 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#556 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
691 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,5,6,7)(9,10,14,19,15,20,12,25)(11,16,23,32,24,33,21,38)(13,18,26,35,29,37,28,17)(22,31,39,34,27,36,40,30);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,20,33)(12,23,14,21)(13,27,29,22)(17,34,35,30)(18,36,37,31)(19,32,25,38)(26,39,28,40);(9,12,15,14)(10,17,20,35)(11,21,24,23)(13,28,29,26)(16,30,33,34)(18,19,37,25)(22,40,27,39)(31,32,36,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37705);