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 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,38)(35,39)(36,40);(2,3)(6,7)(10,13)(11,17)(12,20)(23,24)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(9,11,14,17)(10,15,13,19)(12,21,20,18)(16,24,22,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,16,13,22)(11,18,17,21)(15,23,19,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=h,d^2=f,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×#56903, #8099696, #8100313, … |
| maximal normal subgroup of | C2×#56903, #8099696, #8100313, … |
| maximal quotient of | #586669, #586903, #587309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49891);