Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | Aut(G) = #72285790 (order 2048) · Inn(G) = #1361 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1361 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| prime radical | C2×D16 | C4 |
| 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 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,30,15,32,13,9,16,6,14,8,35,7,33)(3,17,19,41,36,49,20,54,22,50,34,64,21,55,31,48)(4,23,25,45,59,43,61,56,29,57,26,11,28,42,27,44)(10,24,37,53,39,62,46,52,40,58,63,18,47,51,38,60);(1,3)(2,10)(4,24)(5,20)(6,31)(7,34)(8,36)(9,22)(11,41)(12,46)(13,37)(14,38)(15,47)(16,40)(17,43)(18,27)(19,32)(21,30)(23,49)(25,52)(26,60)(28,62)(29,58)(33,63)(35,39)(42,50)(44,48)(45,64)(51,59)(53,61)(54,56)(55,57)Polycyclic presentation:
2^8|a^2=d,c^2=fgh,d^2=efg,e^2=gh,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fgh,g^b=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7447, #150947, #150979, … |
| maximal normal subgroup of | C2×#7447, #150947, #150979, … |
| maximal quotient of | #93862, #93863, #93871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 435);