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 | |
631 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)(34,38)(35,41)(36,44)(46,48);(2,6)(7,12)(8,13)(14,16)(20,28)(23,30)(26,32)(29,31)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(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)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=fg,e^2=g,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×#57031, #8100133, #8100307, … |
| maximal normal subgroup of | C2×#57031, #8100133, #8100307, … |
| maximal quotient of | #585773, #586671, #587229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50019);