Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×#366 |
| prime radical | C23 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,8,13,7,29)(3,14,18,40,16,38,20,51)(4,21,23,52,25,54,24,55)(6,27,26,37,28,56,22,12)(9,30,32,45,31,48,57,17)(10,34,35,44,36,43,53,15)(19,50,49,61,46,64,47,41)(33,42,39,58,59,62,60,63);(1,3,15,42,6,19,23,30,5,18,34,58,26,49,25,45,8,16,44,62,28,46,24,48,7,20,43,63,22,47,4,17)(2,9,21,41,12,33,53,51,29,57,55,64,56,60,36,38,13,31,54,61,37,59,35,40,11,32,52,50,27,39,10,14);(3,16)(9,31)(14,38)(17,45)(18,20)(19,46)(30,48)(32,57)(33,59)(39,60)(40,51)(41,61)(42,62)(47,49)(50,64)(58,63)Polycyclic presentation:
2^8|a^2=e,b^2=df,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #151999, #152422, #152425, … |
| maximal normal subgroup of | #151999, #152422, #152425, … |
| maximal quotient of | #151033, #151053, #151061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6624);