Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
613 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 613 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,11)(8,9)(12,19)(13,14)(15,17)(18,20);(5,6,9,14,10,13,8,11)(7,12,15,19,16,20,17,18);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146596, #146597, #146616, … |
| maximal normal subgroup of | #146596, #146597, #146616, … |
| maximal quotient of | #106094, #106095, #106101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3730);