Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,19,27,20,30,18,12)(8,21,24,36,23,13,22,28)(17,31,34,26,33,29,32,35);(1,2)(3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,25,30)(8,17,22,32,23,33,24,34)(13,26,28,29,21,35,36,31);(2,4)(5,8,9,22,11,23,10,24)(6,13,14,28,15,21,25,36)(7,17,18,32,20,33,19,34)(12,26,16,29,27,35,30,31)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131365, #131371, #131372, … |
| maximal normal subgroup of | #131365, #131371, #131372, … |
| maximal quotient of | #120170, #120171, #120192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4228);