Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator | C22 | C43 |
| socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,10,8,9,11,12);(13,14,15,18,16,17,19,20);(1,2,3,4)(6,9)(10,12)(14,17)(18,20)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,c^a=cg,c^b=ch112 conjugacy classes — view the full character table.
| Fitting subgroup of | #7846⋊C3 |
| maximal subgroup of | #93317, #99290, #99291, … |
| maximal normal subgroup of | #93317, #99290, #99291, … |
| maximal quotient of | #94853, #94857, #94858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 834);