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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(2,6)(7,12)(8,13)(14,16)(20,28)(23,30)(26,32)(29,31)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=fg,d^2=f,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×#54539, #8103684, #8103905, … |
| maximal normal subgroup of | C2×#54539, #8103684, #8103905, … |
| maximal quotient of | #585924, #586934, #587132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47527);