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 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,5)(6,8)(12,20)(15,22)(18,24)(21,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=fgh,d^2=f,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×#56653, #8100074, #8100290, … |
| maximal normal subgroup of | C2×#56653, #8100074, #8100290, … |
| maximal quotient of | #585479, #587135, #587278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49641);