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 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,17)(16,21)(18,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(2,5)(3,7)(10,14)(15,20)(16,21)(22,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,3)(2,6)(4,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(1,2)(3,6)(4,5)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=f,d^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×#58471, #8104369, #8104432, … |
| maximal normal subgroup of | C2×#58471, #8104369, #8104432, … |
| maximal quotient of | #585598, #586548, #586685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51459);