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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,31)(27,30)(29,32);(2,5)(3,7)(17,19,23,21)(18,22,25,26)(20,29,31,27)(24,32,28,30);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,20,23,31)(18,24,25,28)(19,27,21,29)(22,30,26,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,21,23,19)(18,22,25,26)(20,27,31,29)(24,32,28,30);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,15,16,14)(17,22)(18,19)(20,30)(21,25)(23,26)(24,27)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=g,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×#50038, #7705441, #7705683, … |
| maximal normal subgroup of | C2×#50038, #7705441, #7705683, … |
| maximal quotient of | #560883, #560900, #561984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43026);