Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12);(2,4)(13,14)(15,18)(16,17)(19,20);(6,8)(14,17)(18,20);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=h,d^2=g,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| Fitting subgroup of | #13851⋊C3 |
| maximal subgroup of | #107135, #108225, #110604, … |
| maximal normal subgroup of | #107135, #108225, #110604, … |
| maximal quotient of | #153737, C4×#1438, C4×#1446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6839);