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 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32);(2,5)(3,8)(10,12)(11,15)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31);(2,5)(6,7)(9,11,13,15)(10,14,12,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=h,d^2=fgh,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×#58868, #8105074, #8105132, … |
| maximal normal subgroup of | C2×#58868, #8105074, #8105132, … |
| maximal quotient of | #585541, #586018, #586914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51856);