Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C23 |
| commutator | C23 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,8)(9,10,11,12)(13,14,16,15)(18,19);(2,4)(6,8)(14,15)(17,18,20,19);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,16,14)(17,19,20,18)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,c^b=cf88 conjugacy classes — view the full character table.
| Fitting subgroup of | #7556⋊C3 |
| maximal subgroup of | #93557, #97098, #97100, … |
| maximal normal subgroup of | #93557, #97098, #97100, … |
| maximal quotient of | #94849, #123648, #123649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 544);