Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1380 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 160 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 — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(10,16,17,30)(11,21)(12,23)(13,26)(14,27)(18,19,33,34)(20,37,35,32)(22,28)(24,39)(25,38)(29,40,36,31);(2,5)(3,8)(6,7)(9,10,12,18,13,19,24,30,27,35,39,40,22,29,23,32,15,17,25,33,28,34,11,16,14,20,21,31,26,36,38,37)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=cg,d^b=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72227049, #72227122, #72227124, … |
| maximal normal subgroup of | #72227049, #72227122, #72227124, … |
| maximal quotient of | #72227044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1551);