Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C23 | #405 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 170 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:
(9,10,14,19,15,20,29,35,13,18,26,34,28,12,17,27)(11,16,22,31,23,32,37,40,24,21,30,36,38,25,33,39);(1,2,3,5,4,6,7,8)(9,11)(10,16,18,21)(12,25,20,32)(13,24)(14,22)(15,23)(17,33)(19,31,34,36)(26,30)(27,39,35,40)(28,38)(29,37);(9,12,26,35,15,10,17,34,13,20,14,27,28,18,29,19)(11,21,22,36,23,25,37,39,24,16,30,31,38,32,33,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=cg,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216215, #72216232, #72216234, … |
| maximal normal subgroup of | #72216215, #72216232, #72216234, … |
| maximal quotient of | #72225535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36247);