Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29);(1,2)(3,7)(4,5)(6,8)(18,23)(19,26)(25,28)(27,32);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(1,2)(3,6)(4,5)(7,8)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|a^2=fgh,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41636, #7639806, #7640236, … |
| maximal normal subgroup of | C2×#41636, #7639806, #7640236, … |
| maximal quotient of | #544755, #545247, #545791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34624);