Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
743 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(8,9)(12,17)(14,19)(15,18);(1,3,4,2)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #495771, #495774, #495776, … |
| maximal normal subgroup of | #495771, #495774, #495776, … |
| maximal quotient of | #208744, #208752, #208760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14461);