Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151105 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12067 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #508 | C2×C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1313 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,2,6,13,8,15,9,35)(3,16,19,52)(4,22,23,12,25,47,28,48)(5,11,27,43,24,44,29,45)(7,31,30,42,33,63,34,14)(10,37)(17,49)(18,40)(20,60,55,46)(21,38,36,54)(26,50)(32,61,58,53)(39,56)(41,57)(51,59)(62,64);(1,3)(2,10)(4,17)(5,26)(6,21)(7,32)(8,19)(9,36)(11,38)(12,46)(13,49)(14,50)(15,41)(16,43)(18,25)(20,30)(22,61)(23,37)(24,62)(27,56)(28,57)(29,59)(31,51)(33,58)(34,55)(35,40)(39,63)(42,64)(44,54)(45,52)(47,53)(48,60);(1,4,7,5,9,23,34,27,8,25,33,24,6,28,30,29)(2,11,42,22,13,45,63,48,15,44,14,47,35,43,31,12)(3,17,32,26,36,37,55,56,19,18,58,62,21,57,20,59)(10,38,64,61,49,52,39,60,41,54,50,53,40,16,51,46)Polycyclic presentation:
2^9|a^2=e,c^2=efh,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fh,h^a=hi,d^b=dfhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei47 conjugacy classes — view the full character table.
| maximal subgroup of | #72223646, C2×#151105, #72225055, … |
| maximal normal subgroup of | #72223646, C2×#151105, #72225055, … |
| maximal quotient of | #72223644, #72223645, C2×#151105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58301);