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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,12)(14,15)(17,18,21,19)(20,23,24,22);(1,2)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(17,19,21,18)(20,23,24,22);(2,5)(6,7)(9,11,13,16)(10,14,12,15)(17,20)(18,22)(19,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=f,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×#40566, #7639076, #7639574, … |
| maximal normal subgroup of | C2×#40566, #7639076, #7639574, … |
| maximal quotient of | #544447, #545017, #545507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33554);