Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3481 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,7)(6,8)(9,10,13,19,15,18,28,34,14,16,27,30,12,24,26,37)(11,20,22,35,21,17,31,32,23,33,36,40,25,29,38,39);(1,2)(3,8)(4,6)(5,7)(10,16)(13,27)(17,29)(18,24)(20,33)(22,36)(26,28)(31,38);(3,7)(5,8)(9,11)(10,17)(12,25)(13,22)(14,23)(15,21)(16,29)(18,33)(19,32)(20,24)(26,38)(27,36)(28,31)(30,39)(34,40)(35,37)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=egi,f^b=fi,d^c=dgi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36853);