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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(34,38)(35,41)(36,44)(46,48);(2,6)(7,13)(8,11)(14,15)(21,29)(24,31)(26,32)(28,30)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=fg,d^2=g,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×#56231, #8102854, #8103191, … |
| maximal normal subgroup of | C2×#56231, #8102854, #8103191, … |
| maximal quotient of | #585742, #586404, #586745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49219);