Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
329 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,11,12,10)(8,9)(14,17)(15,19)(18,20);(1,2,3,4)(6,9)(10,11);(5,7,8,12)(6,10,9,11)(13,14,16,18)(15,20,19,17)Polycyclic presentation:
2^8|a^2=f,b^2=dgh,c^2=g,e^2=h,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97487, #97489, #97490, … |
| maximal normal subgroup of | #97487, #97489, #97490, … |
| maximal quotient of | #98785, #98786, #98798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1566);