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 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,14,17)(11,13,16,22)(12,15,21,24)(18,20,19,23)(26,30)(27,33)(28,36)(38,40);(2,4)(3,5)(11,16)(13,22)(18,19)(20,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,3,6,5)(2,7,4,8)(9,11,14,16)(10,13,17,22)(12,19,21,18)(15,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,2)(3,7,5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,17)(12,20)(14,22)(15,19)(18,24)(21,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,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×#56996, #8099728, #8100315, … |
| maximal normal subgroup of | C2×#56996, #8099728, #8100315, … |
| maximal quotient of | #585642, #586015, #587189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49984);