Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
359 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(2,4)(13,15,17,25,18,26,16,24)(14,19,20,27,22,28,21,23);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fg112 conjugacy classes — view the full character table.
| maximal subgroup of | #93389, #93390, #93430, … |
| maximal normal subgroup of | #93389, #93390, #93430, … |
| maximal quotient of | #94879, #94893, #94907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 860);