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:
(1,2)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,3)(4,11)(8,10)(12,16)(18,22)(20,29)(23,25)(26,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=f,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×#54675, #8099822, #8100308, … |
| maximal normal subgroup of | C2×#54675, #8099822, #8100308, … |
| maximal quotient of | #585795, #586410, #586751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47663);