Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,19,14,20,31,42,15,18,32,37,34,41,57,60,16,21,33,43,35,44,58,64,12,29,23,48,30,56,49,68)(11,22,25,45,26,46,51,65,27,40,52,62,54,63,69,71,28,47,53,66,55,67,70,72,24,17,36,38,50,39,59,61);(2,5)(6,7)(9,11,23,36,35,55,57,69,15,27,13,25,30,50,58,70,16,28,32,52,14,26,49,59,12,24,33,53,34,54,31,51)(10,17,37,45,44,63,68,72,18,22,43,62,56,67,42,61,21,40,48,66,20,39,60,65,29,47,19,38,41,46,64,71);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60196);