Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150878 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11886 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | #33611 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,5,13,9,15,8,33)(3,16,20,50,23,51,22,59)(4,10,26,35,28,38,29,37)(6,31,30,48,7,32,18,14)(11,39,42,17,43,53,60,21)(12,44,46,27,47,54,36,24)(19,55,57,41,25,40,56,45)(34,61,62,58,49,64,63,52);(2,10)(3,17)(4,24)(5,8)(6,30)(7,18)(11,40)(12,14)(13,35)(15,38)(16,34)(19,56)(20,39)(21,23)(22,53)(25,57)(26,54)(27,28)(29,44)(31,46)(32,36)(33,37)(41,60)(42,45)(43,55)(47,48)(49,51)(50,62)(52,58)(59,63);(1,3,18,39,5,20,6,17,9,23,30,53,8,22,7,21)(2,11,32,59,33,60,48,51,15,43,31,50,13,42,14,16)(4,25,54,64,29,57,27,58,28,19,44,61,26,56,24,52)(10,34,12,45,35,62,46,55,38,49,47,41,37,63,36,40)Polycyclic presentation:
2^9|a^2=e,c^2=ghi,d^2=ghi,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286130, #72287240, #72287275, … |
| maximal normal subgroup of | #72286130, #72287240, #72287275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58079);