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 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(1,2)(3,7)(4,6)(5,8)(10,15)(12,19)(17,21)(20,24)(25,27,30,28)(26,31,34,32)(29,36,38,35)(33,40,37,39);(1,2,4,6)(3,5,8,7)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fh,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×#56517, #8099119, #8099323, … |
| maximal normal subgroup of | C2×#56517, #8099119, #8099323, … |
| maximal quotient of | #586476, #586919, #587047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49505);