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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(11,16)(12,15)(13,21)(17,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fgh,d^2=fg,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×#50577, #7705915, #7706232, … |
| maximal normal subgroup of | C2×#50577, #7705915, #7706232, … |
| maximal quotient of | #560848, #561099, #561936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43565);