Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| 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, prime radical | C24 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 340 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,19,15,20,30,37,14,18,29,32,31,21,33,38)(11,16,23,12,24,26,36,27,22,34,39,25,17,28,35,40);(2,5)(6,7)(9,11,13,23,31,17,33,35,14,22,29,39,15,24,30,36)(10,16,32,12,21,28,37,40,18,34,19,25,20,26,38,27);(1,3,4,8)(2,6,5,7)(9,12,14,25)(10,17,18,24)(11,21,22,20)(13,28,29,26)(15,27,31,40)(16,33,34,30)(19,36,32,35)(23,37,39,38)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dgi,e^b=ei,f^b=fg,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, 49163);