Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
382 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(17,20);(1,3,4,2)(6,9)(7,12)(10,11)(13,14)(15,17)(16,18)(19,20);(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135970, #135971, #135972, … |
| maximal normal subgroup of | #135970, #135971, #135972, … |
| maximal quotient of | #99313, #99319, #99357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1841);