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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,12)(11,15,13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32);(4,8)(6,7)(10,12)(11,13)(18,21)(20,29)(23,25)(26,30);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,19)(18,23)(20,30)(21,25)(22,27)(24,32)(26,29)(28,31);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,12,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=fg,e^2=f,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×#56686, #8099443, #8100102, … |
| maximal normal subgroup of | C2×#56686, #8099443, #8100102, … |
| maximal quotient of | #585643, #585995, #586311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49674);