Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2215 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(13,24)(14,26)(16,30)(17,32)(18,23)(20,37)(21,33)(22,36)(25,28)(27,34)(29,31)(35,40)(38,39);(2,5)(6,8)(9,10,12,15)(11,16,20,29)(13,17,22,27)(14,18,23,26)(19,31,37,30)(21,28,35,39)(24,34,36,32)(25,33,38,40);(1,3,4,7)(2,6,5,8)(9,11,12,20)(10,16,15,29)(13,25,22,38)(14,28,23,39)(17,33,27,40)(18,35,26,21)(19,36,37,24)(30,34,31,32)Polycyclic presentation:
2^8|b^2=h,c^2=d,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13713, #512979, #512981, … |
| maximal normal subgroup of | C2×#13713, #512979, #512981, … |
| maximal quotient of | #152366, #152419, #152501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6701);