Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,20)(11,19,22,37)(12,25,27,32)(13,28,21,18)(14,30,24,16)(17,33,35,23)(26,40,29,38)(31,36,39,34);(1,3,5,4)(2,6,7,8)(9,11,21,14,15,22,13,24)(10,16,28,37,20,30,18,19)(12,23,26,31,27,33,29,39)(17,32,34,40,35,25,36,38);(11,22)(14,24)(16,30)(19,37)(23,33)(25,32)(31,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,15,27)(10,17,20,35)(11,23,22,33)(13,29,21,26)(14,31,24,39)(16,32,30,25)(18,36,28,34)(19,38,37,40)Polycyclic presentation:
2^9|a^2=eh,b^2=efi,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419556);