Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| 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 | |
893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(17,18,21,19)(20,23,24,22);(2,5)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,18)(20,23,24,22);(3,8)(6,7)(10,14)(11,15)(17,20,21,24)(18,22,19,23);(1,3,4,8)(2,6,5,7)(9,11,13,15)(10,12,14,16);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=fh,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 | #131530, #484399, #484455, … |
| maximal normal subgroup of | #131530, #484399, #484455, … |
| maximal quotient of | #544557, #545068, #545427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35009);