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 | |
725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,31)(27,32)(28,29);(1,2,5,6)(3,8,7,4)(9,11,13,12)(10,14,16,15)(18,23)(19,26)(21,30)(27,29);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,26)(18,24,23,31)(20,29,25,27)(21,28,30,32);(1,3,5,7)(2,4,6,8)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28);(1,4)(2,7)(3,6)(5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=f,d^2=h,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×#51147, #7707136, #7707175, … |
| maximal normal subgroup of | C2×#51147, #7707136, #7707175, … |
| maximal quotient of | #560792, #561019, #561883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44135);