Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,6,13,8,12,7,31)(3,14,18,44,17,41,29,45)(4,19,21,40,22,10,30,39)(5,23,26,33,28,55,27,11)(9,32,35,16,37,51,36,24)(15,43,48,63,47,60,49,64)(20,53,54,42,25,56,57,38)(34,59,61,50,52,58,62,46);(1,3,5,24,7,18,27,32,8,17,28,16,6,29,26,51)(2,9,33,14,13,36,55,45,12,37,11,41,31,35,23,44)(4,15,25,46,21,49,57,58,22,47,20,50,30,48,54,59)(10,34,38,43,40,61,56,63,19,52,42,60,39,62,53,64);(1,4)(2,10)(3,15)(5,25)(6,30)(7,21)(8,22)(9,34)(11,42)(12,19)(13,40)(14,43)(16,50)(17,47)(18,49)(20,28)(23,53)(24,46)(26,54)(27,57)(29,48)(31,39)(32,58)(33,38)(35,62)(36,61)(37,52)(41,60)(44,64)(45,63)(51,59)(55,56)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130235, #130236, #130286, … |
| maximal normal subgroup of | #130235, #130236, #130286, … |
| maximal quotient of | #137621, #137661, #137986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6066);