Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #31068 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,20,15,21,32,19)(11,22,25,33,26,39,31,34)(12,18,28,17,29,38,13,30)(16,27,36,23,37,24,35,40);(2,5)(3,8)(6,7)(9,11,15,26)(10,16,21,37)(12,27,29,24)(13,23,28,40)(14,31,32,25)(17,33,30,34)(18,39,38,22)(19,36,20,35);(10,17)(12,28)(13,29)(14,32)(16,33)(18,20)(19,38)(21,30)(22,36)(23,24)(25,31)(27,40)(34,37)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33480);