Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140281 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #468 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2)(3,5)(4,7)(6,8)(9,10,14,19,15,21,33,20)(11,22,24,17,26,35,25,36)(12,27,28,39,30,16,29,34)(13,31,32,18,23,37,40,38);(9,11,23,12,14,25,40,29,15,26,13,30,33,24,32,28)(10,16,18,35,20,34,31,36,21,27,38,22,19,39,37,17);(2,5)(3,8)(6,7)(9,12,13,25,14,29,32,26,15,30,23,24,33,28,40,11)(10,17,18,27,20,35,31,39,21,36,38,16,19,22,37,34)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=eghi,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72309417, #72309427, #72309428, … |
| maximal normal subgroup of | #72309417, #72309427, #72309428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47478);