Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,12,17,13,18,23,31,14,15,22,26,24,27,36,38)(11,16,20,29,19,30,32,40,21,25,35,37,33,28,34,39);(1,2,4,3,5,6,8,7)(10,15)(11,19,21,33)(12,22)(16,28,25,30)(18,27)(20,34,35,32)(23,36)(29,40,37,39);(1,3,4,6,5,7,8,2)(9,11,12,20,13,19,23,32,14,21,22,35,24,33,36,34)(10,16,17,29,18,30,31,40,15,25,26,37,27,28,38,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch,d^a=di,c^b=cg,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35486);