Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72317497 (order 2048) · Inn(G) = C42 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×#346 |
| prime radical | C23 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,5,11,19,31,6,12,20,32,22,35,46,57,7,10,21,30,23,33,47,55,24,34,48,56,49,58,63,64)(3,13,15,37,16,38,41,60,17,26,42,29,43,52,61,54,14,36,18,44,39,59,45,62,40,8,25,27,50,28,51,53);(1,3)(2,8,10,26)(4,18)(5,16)(6,17)(7,14)(9,29,30,27)(11,28,33,52)(12,13,34,36)(15,21)(19,45)(20,25)(22,43)(23,39)(24,40)(31,54,55,53)(32,44,56,37)(35,38,58,59)(41,47)(42,48)(46,51)(49,50)(57,62,64,60)(61,63);(3,14)(8,26)(13,36)(15,18)(16,39)(17,40)(25,42)(27,29)(28,52)(37,44)(38,59)(41,45)(43,50)(51,61)(53,54)(60,62)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,b^a=bg,e^a=eh,c^b=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #150163, #150164, #150165, … |
| maximal normal subgroup of | #150163, #150164, #150165, … |
| maximal quotient of | #149496, #149501, #149508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6539);