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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,17)(11,13,16,22)(12,15,19,23)(18,21,20,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(11,16)(12,19)(13,22)(15,23)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,13,17,22)(12,20,19,18)(15,24,23,21)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,21)(14,19)(16,20)(17,23)(22,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,13)(10,16)(11,17)(12,21)(14,22)(15,20)(18,23)(19,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=fg,e^2=f,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×#50213, #7706177, #7706214, … |
| maximal normal subgroup of | C2×#50213, #7706177, #7706214, … |
| maximal quotient of | #561232, #561394, #562427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43201);