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 | C4⋊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 | |
156 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,12,18,13,20,24,33,14,16,25,28,26,29,37,38,15,19,23,32,27,34,36,40,11,17,21,30,22,31,35,39);(1,2,4,7,5,8,3,6)(10,16,19,17)(12,23)(18,30,32,28)(20,29,34,31)(21,25)(24,36)(33,39,40,38)(35,37)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,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 | #72225413, #72225513, #72225531, … |
| maximal normal subgroup of | #72225413, #72225513, #72225531, … |
| maximal quotient of | #72225409, #72227190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1212);