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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,2,5,6)(3,8,7,4)(11,18)(14,17)(19,24)(21,23)(26,31)(28,35)(33,37)(36,40);(2,6)(3,7)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,27,30,32)(26,28,31,35)(29,34,38,39)(33,36,37,40);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fh,d^2=gh,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×#58861, #8105127, #8105132, … |
| maximal normal subgroup of | C2×#58861, #8105127, #8105132, … |
| maximal quotient of | #585356, #586050, #587074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51849);