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 | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
823 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30);(1,2,4,6)(3,5,8,7)(10,14)(12,15)(18,21)(19,25)(24,29)(26,28);(9,11)(10,12)(13,16)(14,15)(17,19,22,25)(18,23,21,27)(20,28,30,26)(24,32,29,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,30)(18,24,21,29)(19,26,25,28)(23,31,27,32);(1,3,4,8)(2,5,6,7)(9,11)(10,12)(13,16)(14,15)(17,21,22,18)(19,27,25,23)(20,29,30,24)(26,32,28,31)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=h,d^2=h,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×#51500, #7707338, #7707348, … |
| maximal normal subgroup of | C2×#51500, #7707338, #7707348, … |
| maximal quotient of | #560777, #560948, #561893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44488);