Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72317567 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C22×C4×D4 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22⋊C4 |
| commutator | C22 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×C22⋊C16 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,10,5,11,19,32,6,12,20,33,22,34,44,56,7,9,21,28,23,29,45,51,24,30,46,52,47,53,61,63)(3,13,15,31,16,35,39,54,17,36,40,55,41,57,59,64,14,8,18,25,37,26,42,48,38,27,43,49,58,50,60,62);(1,3)(2,8)(4,18)(5,16)(6,17)(7,14)(9,13)(10,31)(11,26)(12,27)(15,21)(19,42)(20,43)(22,41)(23,37)(24,38)(25,28)(29,35)(30,36)(32,54)(33,55)(34,50)(39,45)(40,46)(44,60)(47,58)(48,51)(49,52)(53,57)(56,64)(59,61)(62,63);(3,14)(8,13)(15,18)(16,37)(17,38)(25,31)(26,35)(27,36)(39,42)(40,43)(41,58)(48,54)(49,55)(50,57)(59,60)(62,64)Polycyclic presentation:
2^8|a^2=e,e^2=f,f^2=g,g^2=h,b^a=bd,d^a=dh,c^b=ch,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #94126, #94127, #149669, … |
| maximal normal subgroup of | #94126, #94127, #149669, … |
| maximal quotient of | #149519, #149601, #149655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6544);