Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #95 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C22⋊C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 2 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,13,18,14,11,21,23,12,17,26,33,27,22,32,37,15,20,29,19,30,25,36,24,28,35,39,34,40,38,16,31);(1,2,4,7,5,8,3,6)(9,11,13,24,14,17,21,34,12,22,26,31,27,20,32,18,15,25,29,23,30,35,36,33,28,38,39,37,40,10,16,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fg,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227023, #72227124, #72227133, … |
| maximal normal subgroup of | #72227023, #72227124, #72227133, … |
| maximal quotient of | #72227004 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1233);