Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8)(10,12)(14,15);(5,6)(7,8)(13,14,16,15);(9,10,11,12)(13,15,16,14);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,b^a=be,c^a=cf,d^a=dg88 conjugacy classes — view the full character table.
| maximal subgroup of | #99735, #99738, #99751, … |
| maximal normal subgroup of | #99735, #99738, #99751, … |
| maximal quotient of | #154079, #154088, #154089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7686);