Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,29)(27,31,35,34)(28,36,38,32)(33,40,37,39);(1,2,5,4)(3,6,7,8)(10,15)(11,16)(17,22)(18,23)(26,29)(31,34)(32,36)(39,40);(1,3,5,7)(2,6,4,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,32)(27,33)(29,36)(30,38)(31,39)(34,40)(35,37);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,26)(27,34,35,31)(28,36,38,32)(33,40,37,39)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41577, #7639971, #7640233, … |
| maximal normal subgroup of | C2×#41577, #7639971, #7640233, … |
| maximal quotient of | #544721, #545208, #545720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34565);