Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,17)(12,14)(13,20)(15,19)(16,21)(18,22)(23,24)(25,26,31,33)(27,34,28,29)(30,38,39,32)(35,37,36,40);(2,7)(3,4)(10,12)(13,21)(15,17)(18,24)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(1,3,6,4)(2,8,7,5)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,4,6,3)(2,5,7,8)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(1,5)(2,3)(4,7)(6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,32,33,38)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54695, #8103601, #8103696, … |
| maximal normal subgroup of | C2×#54695, #8103601, #8103696, … |
| maximal quotient of | #585731, #586346, #586754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47683);