Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 143 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)(6,9)(7,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(5,6,7,10,8,9,11,12)(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,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94326, #105451, #105452, … |
| maximal normal subgroup of | #94326, #105451, #105452, … |
| maximal quotient of | #95224, #95226, #95230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1026);