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 | C22×D4 | C23 |
| 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)(3,8)(4,7)(5,6)(10,14)(11,17)(16,21)(18,23)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,3)(2,7)(4,6)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(3,4)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,4,6,3)(2,7,5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,5,6,2)(3,7,4,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=fgh,c^2=g,d^2=h,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×#58517, #8104264, #8104434, … |
| maximal normal subgroup of | C2×#58517, #8104264, #8104434, … |
| maximal quotient of | #585443, #586612, #586677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51505);