Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #72283959 (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator, prime radical | C23 | C2×C16 |
| Frattini, cosocle | C2×M32 | C22 |
| socle | C2 | #1322 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,5,11,18,31,6,12,19,32,20,33,42,54,7,9,17,27,21,28,40,49,22,29,41,50,43,51,59,62)(3,8,14,23,15,25,37,44,16,26,38,45,39,48,58,60,13,24,34,30,35,46,55,52,36,47,56,53,57,61,64,63);(1,3)(2,8,9,24)(4,14)(5,15)(6,16)(7,13)(10,30,27,23)(11,25,28,46)(12,26,29,47)(17,34)(18,37)(19,38)(20,39)(21,35)(22,36)(31,52,49,44)(32,53,50,45)(33,48,51,61)(40,55)(41,56)(42,58)(43,57)(54,63,62,60)(59,64)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=ch,d^b=de,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93713, #93714, #93741, … |
| maximal normal subgroup of | #93713, #93714, #93741, … |
| maximal quotient of | #93718, #93724, #93730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 353);