Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C23 | #418 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 3 generators on 40 points:
(9,10,13,19)(11,23)(12,20,29,37)(14,31,30,18)(15,22,16,21)(17,32)(24,34)(25,36)(26,33)(27,38)(28,39)(35,40);(1,2,3,5,4,6,7,8)(9,11,14,26,16,24,29,32,13,28,30,40,15,27,12,25)(10,17,20,34,22,33,18,23,19,36,37,38,21,35,31,39);(11,24,28,27)(12,29)(14,30)(17,33,36,35)(18,31)(20,37)(23,34,39,38)(25,40,32,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309752, #72309759, #72309761, … |
| maximal normal subgroup of | #72309752, #72309759, #72309761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35752);