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 | |
691 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,4,6)(3,5,8,7)(10,14)(11,18)(16,20)(19,24)(28,36)(31,38)(34,40)(37,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=gh,d^2=h,e^2=g,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×#58772, #8104959, #8104997, … |
| maximal normal subgroup of | C2×#58772, #8104959, #8104997, … |
| maximal quotient of | #585888, #586381, #586861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51760);