Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13493 |
| prime radical | C24 | Q32 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,19,31,44,40,6,12,21,34,25,35,51,59,7,10,22,32,49,58,64,63,24,36,50,60,5,11,20,33)(3,13,15,37,23,53,48,29,17,41,45,56,43,57,46,27,14,39,42,62,54,61,52,38,26,55,47,28,16,30,18,8);(8,27)(13,39)(28,56)(29,38)(30,57)(37,62)(41,55)(53,61);(1,3,6,26,7,14,24,17)(2,8,12,38,10,27,36,29)(4,18,21,52,22,46,50,48)(5,23,19,16,25,54,49,43)(9,30,34,61,32,57,60,53)(11,37,31,28,35,62,58,56)(13,40,55,59,39,63,41,33)(15,44,47,51,42,64,45,20)Polycyclic presentation:
2^9|a^2=e,c^2=dh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^c=di,e^c=eg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320880, #72320895, #72320903, … |
| maximal normal subgroup of | #72320880, #72320895, #72320903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60568);