Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26)(27,32)(28,29)(30,31)(33,38)(34,39)(35,40)(36,37);(1,2,5,4)(3,6,7,8)(10,15)(12,19)(17,21)(20,24)(26,31)(27,34)(29,38)(35,37);(1,3)(2,6)(4,8)(5,7)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,27,30,34)(26,32,31,39)(28,37,33,35)(29,36,38,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50968, #7706610, #7706670, … |
| maximal normal subgroup of | C2×#50968, #7706610, #7706670, … |
| maximal quotient of | #560856, #561146, #561943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43956);