Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | Aut(G) = #72286063 (order 2048) · Inn(G) = #1365 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1365 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| prime radical | C22 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,30,14,25,11,6,33,31,43,36,42,28,62,9,15,26,47,7,34,27,44,8,13,32,46,24,49,4,23)(3,16,18,38,57,41,56,50,19,45,58,10,35,37,17,39,22,53,60,48,20,61,55,51,21,52,59,40,54,64,29,63);(1,3)(2,10,15,40)(4,18)(5,29)(6,21)(7,35)(8,19)(9,22)(11,41,44,61)(12,45,47,52)(13,48,33,38)(14,50,34,51)(16,46,53,43)(17,26)(20,36)(23,37,62,64)(24,57)(25,59)(27,58)(28,60)(30,54)(31,55)(32,56)(39,42,63,49)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=fgh,d^2=efgh,e^2=h,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=def,e^b=eh,f^b=fg,g^b=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7454, #150964, #150979, … |
| maximal normal subgroup of | C2×#7454, #150964, #150979, … |
| maximal quotient of | #93940, #94363, C2×#7454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 442);