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 | #392 | C22 |
| 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:
(2,4)(6,7)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26)(27,33)(28,35)(29,37)(30,31)(32,38)(34,36)(39,40);(1,2,5,4)(3,7,8,6)(10,12)(11,17)(13,21)(22,24)(26,31)(28,33)(29,38)(34,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=g,d^2=fg,e^2=f,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×#55903, #8103087, #8103186, … |
| maximal normal subgroup of | C2×#55903, #8103087, #8103186, … |
| maximal quotient of | #586013, #586517, #586803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48891);