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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,26,27,23)(20,24,30,21)(28,32,31,29);(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,16,15)(19,27)(20,30)(21,24)(23,26);(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,31)(21,29)(22,27)(24,32)(25,26)(28,30);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,28)(21,25)(22,30)(23,29)(26,32)(27,31);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,21,22,24)(18,20,25,30)(19,29,27,32)(23,31,26,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fgh,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×#50998, #7706547, #7706722, … |
| maximal normal subgroup of | C2×#50998, #7706547, #7706722, … |
| maximal quotient of | #560824, #561131, #561912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43986);