Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,6,7,8)(9,10,14,19,16,23,31,20,17,21,12,30,15,22,13,32)(11,24,25,18,29,35,26,37,27,39,40,38,28,34,33,36);(1,2,3,6,5,7,4,8)(9,11,15,29,17,27,16,28)(10,18,23,36,21,38,22,37)(12,26,31,40,14,33,13,25)(19,24,20,34,30,39,32,35);(1,3,5,4)(2,6,7,8)(9,12,15,31,17,14,16,13)(10,19,22,32,21,30,23,20)(11,25,28,33,27,40,29,26)(18,34,36,39,38,35,37,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43841);