Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404632 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,16,12,15,20,14,19)(11,21,24,17,26,27,25,22)(13,28,31,34,30,33,29,18)(23,32,37,40,39,36,38,35);(1,3)(2,5)(4,7)(6,8)(9,11,16,24,15,26,14,25)(10,17,12,27,20,22,19,21)(13,23,31,37,30,39,29,38)(18,32,28,40,34,36,33,35);(3,7)(5,8)(9,12,14,20,15,19,16,10)(11,22,24,27,26,17,25,21)(13,28,29,34,30,33,31,18)(23,36,37,40,39,32,38,35);(9,13,16,31,15,30,14,29)(10,18,19,33,20,34,12,28)(11,23,24,37,26,39,25,38)(17,32,21,35,22,36,27,40)Polycyclic presentation:
2^9|a^2=efgh,b^2=ef,c^2=e,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322724);