Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #3169 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,3,7,5,8,4,6)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,12,22,25,21,26,32,39,23,27,34,40,38,24,16,31);(2,6)(3,4)(7,8)(9,11,14,22,13,21,28,32,15,23,30,34,29,38,33,16)(10,12,18,25,19,26,36,39,20,27,37,40,17,24,35,31);(2,7)(3,4)(6,8)(9,12)(10,16)(11,18)(13,26)(14,25)(15,27)(17,34)(19,22)(20,32)(21,36)(23,37)(24,29)(28,39)(30,40)(31,33)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egi,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43865);