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)(9,10,14,18,15,19,33,22,13,20,30,26,31,35,40,38)(11,21,23,27,25,36,39,37,24,16,32,12,17,28,34,29);(1,2,4,3,5,6,8,7)(9,11,13,24)(10,16)(12,26)(14,32,30,23)(15,25,31,17)(18,27)(19,28)(20,21)(22,37)(29,38)(33,34,40,39)(35,36);(1,3,4,6,5,7,8,2)(9,12,14,28,15,29,33,21,13,27,30,36,31,37,40,16)(10,17,18,34,19,11,22,23,20,25,26,39,35,24,38,32)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,f^a=fi,c^b=cg,d^b=di,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, 35487);