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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,14)(11,12,15,16)(19,23)(21,24);(1,2)(3,4)(5,6)(7,8)(10,14)(11,15)(17,18,20,22)(19,21,23,24);(2,6)(3,7)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,21,22,24);(1,3,5,7)(2,4,6,8)(9,12)(10,11)(13,16)(14,15);(1,4)(2,7)(3,6)(5,8)Polycyclic presentation:
2^8|a^2=f,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 | C2×#42010, #7640316, #7640431, … |
| maximal normal subgroup of | C2×#42010, #7640316, #7640431, … |
| maximal quotient of | #544507, #545103, #545360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34998);