Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| 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)(11,21)(12,25)(13,23)(14,24)(17,18)(19,36)(20,22)(26,39)(27,37)(28,38)(29,30)(31,35)(32,34)(33,40);(2,6,8,7)(9,10,12,17,27,32,11,22,24,35,39,30,23,29,38,31,15,20,28,34,13,18,26,16,14,19,21,33,37,40,25,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72223668, #72223674, C8×#1419, … |
| maximal normal subgroup of | #72223668, #72223674, C8×#1419, … |
| maximal quotient of | C2×#94049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1245);