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 | |
758 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)(3,4)(10,13)(11,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(1,2,6,5)(3,8,4,7)(10,13)(14,16)(20,28)(23,30)(26,32)(29,31);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,5,6,2)(3,8,4,7)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fgh,d^2=f,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×#56652, #8100043, #8100290, … |
| maximal normal subgroup of | C2×#56652, #8100043, #8100290, … |
| maximal quotient of | #585479, #586111, #586254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49640);