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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,31)(27,32,35,38)(28,33,37,40)(29,36,39,34);(1,2,4,6)(3,5,8,7)(10,14)(15,21)(16,19)(22,23)(26,31)(27,35)(28,37)(29,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,34)(29,33)(30,35)(31,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,28)(26,33)(27,36)(29,38)(30,37)(31,40)(32,39)(34,35);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,29,30,39)(26,34,31,36)(27,33,35,40)(28,38,37,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fgh,e^2=gh,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×#58964, #8105238, #8105250, … |
| maximal normal subgroup of | C2×#58964, #8105238, #8105250, … |
| maximal quotient of | #585254, #586112, #586150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51952);