Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(2,5)(3,7)(9,11,13,10)(12,15,16,14)(18,22)(23,29)(24,27)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,3,4,7)(2,6,5,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(1,2)(3,6)(4,5)(7,8)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46021, #7702643, #7702707, … |
| maximal normal subgroup of | C2×#46021, #7702643, #7702707, … |
| maximal quotient of | #557087, #557111, #557371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39009);