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:
(2,4)(6,7)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,30,33)(27,29,32,38)(28,31,37,40)(34,36,35,39);(1,2,5,4)(3,7,8,6)(10,14)(11,18)(16,20)(19,24)(27,32)(29,38)(34,35)(36,39);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27,30,32)(26,29,33,38)(28,35,37,34)(31,39,40,36);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,32)(27,33)(28,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=h,d^2=fg,e^2=fg,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×#57067, #8099538, #8100308, … |
| maximal normal subgroup of | C2×#57067, #8099538, #8100308, … |
| maximal quotient of | #586560, #586722, #587022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50055);