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 | #2825 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(10,13)(11,15)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30);(1,2,4,5)(3,8,7,6)(10,13)(14,16)(18,20)(23,25)(24,28)(30,31);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,3,4,7)(2,6,5,8)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=f,d^2=fgh,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×#58394, #8104336, #8104352, … |
| maximal normal subgroup of | C2×#58394, #8104336, #8104352, … |
| maximal quotient of | #585484, #585908, #586750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51382);