Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
319 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 135 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)(5,6,10,11)(7,9,12,8)(13,14,17,20,18,21,16,26)(15,22,23,28,25,27,24,19);(2,4)(5,7,9,11,10,12,8,6)(13,15)(14,19,21,28)(16,24)(17,23)(18,25)(20,22,26,27);(5,8,10,9)(6,7,11,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,27,28,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105883, #106101, #107244, … |
| maximal normal subgroup of | #105883, #106101, #107244, … |
| maximal quotient of | #94884, #94895, #94899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 888);