Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(2,7)(3,8)(4,5)(9,10,13,18,22,36,28,17,14,19,29,37,15,20,12,27)(11,21,25,16,31,33,24,38,23,39,40,32,26,35,30,34);(1,2,5,3,6,8,4,7)(9,11,22,26,14,23,15,31)(10,16,20,38,19,32,36,34)(12,25,13,30,28,40,29,24)(17,35,18,21,27,33,37,39);(1,3,5,8,6,7,4,2)(11,23)(16,32)(21,39)(24,30)(25,40)(26,31)(33,35)(34,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=e,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,h^b=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309716, #72309720, #72309738, … |
| maximal normal subgroup of | #72309716, #72309720, #72309738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48491);