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 | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,12)(13,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(2,5)(3,7)(11,13)(12,15)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31);(1,2,4,5)(3,6,7,8)(9,11,14,13)(10,15,16,12)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,15)(10,11,16,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,15,16,12)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=fg,e^2=g,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×#50009, #7705231, #7705682, … |
| maximal normal subgroup of | C2×#50009, #7705231, #7705682, … |
| maximal quotient of | #560834, #561036, #561954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42997);