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 | |
716 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)(3,5)(4,7)(6,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,3)(4,8)(13,21)(16,23)(18,24)(20,22)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(2,3)(6,7)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=h,c^2=fg,d^2=g,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×#56232, #8102877, #8103191, … |
| maximal normal subgroup of | C2×#56232, #8102877, #8103191, … |
| maximal quotient of | #585324, #586745, #586766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49220);