Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #367 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C23 | #358 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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 84 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,4,7,5,8,3,6)(10,16,17,32)(11,21)(12,26)(13,29)(14,23)(18,37,36,19)(20,40,33,35)(22,30)(24,28)(25,27)(31,38,39,34);(2,6,8,7)(9,10,12,18,13,19,11,16,23,33,21,34,22,31,26,40,15,17,28,36,30,37,25,32,14,20,27,38,29,39,24,35)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72223668, C8×#1426, #72223677, … |
| maximal normal subgroup of | #72223668, C8×#1426, #72223677, … |
| maximal quotient of | C2×#94046 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1242);