Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,20,15,19,17,31,16,11,21,25,13,24,22,32)(12,23,27,37,26,34,38,39,28,18,30,35,29,36,33,40);(2,5)(6,7)(9,11,22,31,15,24,14,25,16,10,17,32,13,19,21,20)(12,23,38,39,26,34,30,35,28,18,33,40,29,36,27,37);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,23)(13,29)(14,30)(15,26)(16,28)(17,33)(19,36)(20,37)(21,27)(22,38)(24,34)(25,35)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=di,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49162);