Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
729 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,21,22)(19,25,26,23)(20,24,29,30)(27,32,28,31);(2,6)(4,7)(9,11,13,12)(10,14,16,15)(18,22)(19,26)(24,30)(27,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,26)(18,23,22,25)(20,28,29,27)(24,32,30,31);(1,4,5,7)(2,3,6,8)(17,20,21,29)(18,24,22,30)(19,27,26,28)(23,31,25,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50421, #7706177, #7706227, … |
| maximal normal subgroup of | C2×#50421, #7706177, #7706227, … |
| maximal quotient of | #560762, #560936, #561845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43409);