Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
792 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)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30);(1,2,4,6)(3,5,7,8)(19,25)(21,29)(23,27)(24,28);(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32);(1,2)(3,5)(4,6)(7,8)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44859, #7669282, #7669288, … |
| maximal normal subgroup of | C2×#44859, #7669282, #7669288, … |
| maximal quotient of | #535799, #535800, #536454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37847);